Notes related to “Operators in quantum mechanics”
Armin Scrinzi
July 11, 2017
USE WITH CAUTION
These notes are compilation of my “scribbles” (only SCRIBBLES, although typeset in LaTeX). There
absolutely no time to unify notation, correct errors, proof-read, and the like. Your primary source must
by your own notes.
For now, it is only an outline to list a few of the topics that will be covered. The notes will be partially
be filled in as the lecture proceeds.
Contents summary
The lecture consist of three main parts:
• the abstract motivation of quantum mechanics (C* algebras, GNS construction),
• key elements of the theory of unbounded operators (spectral theorem, Stone theorem), and
• elements of scattering theory (Møller operators, proof of asymptotic completeness).
Essential functional analysis will be introduced and selected topics will be treated in full mathematical
rigor. The course is intended for physicist with strong mathematical inclinations, but contact to the
epistemological concepts of physics will be sought.
Purpose of the lecture
Quantum physics has spawned extensive use of functional analysis, but students in their majority are
left with little basic knowledge of the subject. A solid foundation of many techniques that are blindly
used by most physicists typically takes a 2-3 semester course in mathematics, which still may miss some
advanced issues that are relevant in practice. In addition, in a mathematics course usually little attention
is paid to the connection to the original motivation, the formation of intuitive pictures, and the actual
importance of certain formal questions.
This lecture is intended for physicist with strong mathematical inclinations. In a few points, concepts
will be introduced with mathematical rigor, which should put you into the position to at least see the
major inaccuracies committed in the remaining topics. In this spirit, an overview of many mathematical
questions arising in quantum mechanics will be given and at least sketches of solutions and ideas of proofs
will be provided.
Literature
The course of the lecture is strongly inspired by
W. Thirring - Lehrbuch der Mathematischen Physik
1
The part on unbounded operators follows closely
G. Teschl - Mathematical Methods in Quantum Mechanics
For basic reference we will largely use
Reed/Simon - Methods of Modern Mathematical Physics
Further literature will we given for selected topics.
2
Contents
1 The
1.1
1.2
1.3
1.4
C ∗ -algebra approach to classical and quantum physics
An analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Observables . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 States . . . . . . . . . . . . . . . . . . . . . . . . . . .
Justification of the C ∗ algebraic nature of observables . . . .
1.4.1 Observables . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 State . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Normalization and positivity of a state . . . . . . . . .
1.4.4 Equality of observables . . . . . . . . . . . . . . . . .
1.4.5 C ∗ -structure . . . . . . . . . . . . . . . . . . . . . . .
1.4.6 Basic structure of a physical theory . . . . . . . . . .
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8
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12
2 C ∗ algebra
2.1 Banach space . . . . . . . . . . . . . . . . . . . .
2.1.1 Comment . . . . . . . . . . . . . . . . . .
2.2 Classes of elements of a C ∗ algebra . . . . . . . .
2.3 Open set . . . . . . . . . . . . . . . . . . . . . . .
2.4 The spectrum of A ∈ A . . . . . . . . . . . . . .
2.4.1 Series expansion of the resolvent . . . . .
2.4.2 The resolvent set is an open set . . . . . .
2.4.3 Algebra element properties and spectrum
2.5 Problems . . . . . . . . . . . . . . . . . . . . . .
2.6 State on a C ∗ algebra . . . . . . . . . . . . . . .
2.6.1 Notes . . . . . . . . . . . . . . . . . . . .
2.6.2 The states form a convex set . . . . . . .
2.6.3 Pure and mixed states . . . . . . . . . . .
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13
13
15
15
15
15
16
16
16
16
20
20
20
20
3 Gelfand isomorphism
3.1 Weak *-topology . . . . . . . . . . . .
3.2 Gelfand isomorphism . . . . . . . . . .
3.2.1 Terms: Closure and dense sets
3.2.2 The spectral theorem in nuce .
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22
23
24
25
25
4 Representations in Hilbert space
4.1 Operator norm . . . . . . . . . . . . . . . .
4.2 Representation of an algebra . . . . . . . .
4.2.1 Note . . . . . . . . . . . . . . . . . .
4.3 Irreducible representation . . . . . . . . . .
4.4 Sum and tensor product of representations .
4.4.1 Illustration . . . . . . . . . . . . . .
4.5 Illustration . . . . . . . . . . . . . . . . . .
4.6 Problems . . . . . . . . . . . . . . . . . . .
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26
26
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27
27
28
28
28
. . . .
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irreps
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29
30
30
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31
31
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5 The GNS construction
5.1 Reducible and irreducible GNS representations . . . . . .
5.1.1 1 ∈ A/N is cyclic . . . . . . . . . . . . . . . . . . .
5.1.2 Irrep ⇔ pure state . . . . . . . . . . . . . . . . . .
5.1.3 Irreps for abelian C ∗ algebras . . . . . . . . . . . .
5.1.4 Any representation is the (possibly infinite) sum of
3
6 Towards the spectral theorem for bounded operators
33
7 Elements of measure theory
7.1 Σ-algebra . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Measure and measurable space . . . . . . . . . . . . .
7.4 A few definitions for the Lebesgue integral (reminder)
7.5 Lebesgue measure . . . . . . . . . . . . . . . . . . . .
7.6 Absolutely continuous . . . . . . . . . . . . . . . . . .
7.6.1 Examples . . . . . . . . . . . . . . . . . . . . .
7.7 Signed measures and complex valued measures . . . .
7.8 Projector valued measures . . . . . . . . . . . . . . . .
7.9 Positive operator valued measures . . . . . . . . . . .
7.10 Measures and continuous functionals . . . . . . . . . .
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33
33
33
34
34
35
35
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36
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37
37
8 The
8.1
8.2
8.3
8.4
8.5
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37
38
38
38
38
39
9 Bounded and unbounded operators in Hilbert space
9.1 Linear operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 B.L.T. theorem (bounded line transformation) . . . . . . . . . .
9.2.1 Multiplication operator on L2 (dx, R) . . . . . . . . . . . .
9.3 Comments on the domain . . . . . . . . . . . . . . . . . . . . . .
9.4 Quadratic form and polarization identity . . . . . . . . . . . . . .
9.4.1 Hermitian operator . . . . . . . . . . . . . . . . . . . . . .
9.5 Adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.1 Comment on the domain D(A∗ ) . . . . . . . . . . . . . .
9.6 Range and kernel . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 Self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.1 Multiplication operator . . . . . . . . . . . . . . . . . . .
9.8 Closed operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9 Normal operator . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9.1 The domain of self-adjoint operators is maximal . . . . .
9.9.2 Closure of a symmetric operator . . . . . . . . . . . . . .
9.9.3 The importance of self-adjointness for quantum mechanics
9.9.4 Example: the spatial derivative . . . . . . . . . . . . . . .
9.9.5 A criterion for self-adjointness . . . . . . . . . . . . . . . .
9.9.6 Spectrum of a self-adjoint operator . . . . . . . . . . . . .
9.10 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . .
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41
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41
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45
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46
46
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48
48
49
spectral theorem for self-adjoint operators
Bounded spectral theorem with projection valued measure on R . .
Resolution of identity P (λ) . . . . . . . . . . . . . . . . . . . . . .
Construction of operators from projection valued measures . . . .
10.3.1 The subspace Hψ . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Unitarity, cyclic vectors, spectral basis . . . . . . . . . . . .
10.4 For every projection valued measure there is a self-adjoint operator
10.5 For every self-adjoint operator there is a spectral measure . . . . .
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50
50
51
51
53
53
54
54
spectral theorem for normal operators
Gel’fand isomorphism . . . . . . . . . . . . . . .
A state defines a measure on σ(A) . . . . . . . .
Construction of the L2 (dµi , σ(A)) . . . . . . . . .
Spectral theorem by projection valued measures .
Quantum habits . . . . . . . . . . . . . . . . . .
10 The
10.1
10.2
10.3
4
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10.5.1
10.5.2
10.5.3
10.5.4
10.5.5
10.5.6
The resolvent of P (a) . . . . . . . . . . .
A measure is uniquely defined by its Borel
The resolvent defines a measure dµψ . . .
A resolution of identity P (λ) . . . . . . .
The spectral projectors . . . . . . . . . .
Spectral theorem for self-adjoint operators
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transform
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54
54
55
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56
11 Stone’s theorem
11.1 A self-adjoint operator generates a unitary group . . . . . . . . . . . . . . . . . . . . . . .
11.2 Stone theorem - unitary groups derive from self-adjoint operators . . . . . . . . . . . . . .
56
56
57
12 Further important topics
12.1 Self-adjoint Hamiltonians . . . . . . . . . . . . . .
12.2 Self-adjointness of typical Hamiltonians in Physics
12.2.1 Domain of essential self-adjointness . . . . .
12.2.2 Self-adjoint −∆ + V . . . . . . . . . . . . .
12.3 Perturbation theory . . . . . . . . . . . . . . . . .
12.4 Analyticity of eigenvalues . . . . . . . . . . . . . .
12.4.1 Riesz projections . . . . . . . . . . . . . . .
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59
59
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61
13 Scattering theory
13.1 Scattering is our window into the microscopic world . . . . . . . . . . . . .
13.1.1 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.2 Potential scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Mathematical modeling of a (simple) scattering experiment . . . . . . . . .
13.3 The Møller operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 The Lippmann-Schwinger (LS) equation . . . . . . . . . . . . . . . . . . . .
13.4.1 In- and Out-states . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5 Spectral properties and dynamical behavior . . . . . . . . . . . . . . . . . .
13.5.1 Spectral subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.2 Fourier transform of smooth functions (Riemann-Lebesgue) . . . . .
13.5.3 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.4 The RAGE theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6 The Møller operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.7 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.7.1 Asymptotic completeness . . . . . . . . . . . . . . . . . . . . . . . .
13.7.2 Scattering operators compare two dynamics . . . . . . . . . . . . . .
13.7.3 Intertwining property . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8 Asymptotic completeness - guide through the proof by Enss . . . . . . . . .
13.8.1 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8.2 Exclusion of the sing spectrum . . . . . . . . . . . . . . . . . . . . .
13.8.3 In- and outgoing spaces . . . . . . . . . . . . . . . . . . . . . . . . .
13.8.4 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8.5 Schwartz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8.6 Compact operators, weak convergence . . . . . . . . . . . . . . . . .
13.8.7 Short range potentials . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8.8 Escape of the wave packet - Perry’s estimate . . . . . . . . . . . . .
13.8.9 Existence of scattering operators for short range potentials . . . . .
13.8.10 Compactness — in- and outgoing parts are almost free wave packets
13.8.11 Final steps of the proof . . . . . . . . . . . . . . . . . . . . . . . . .
13.9 Coulomb scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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64
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13.9.1 Exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9.2 Coulomb + finite range potentials . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Formal derivations
14.1 The Lippmann-Schwinger equation in disguise .
14.2 Ω± in the spectral representation of H0 . . . .
14.2.1 Remark on “ac-spectral eigenfunctions”
14.2.2 The LS equation in position space . . .
14.3 Mapping free to free spaces: the S-matrix . . .
14.3.1 Spectral representation of the S-matrix
14.3.2 The T -matrix . . . . . . . . . . . . . . .
14.4 Scattering cross section — derivation . . . . . .
14.4.1 Post and prior form . . . . . . . . . . .
6
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84
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88
Preliminary remarks
I was remarked that students are so overwhelmed by the formalism of quantum mechanics that they have
no chance to think what it implied. The just have to “shut up and calculate”.
I absolutely agree with that statement. If it were not for practical necessity, one should ban the
way that quantum mechanics is (and inevitably must be) taught in a first course. In its epistemological
content it is often communicated by indoctrination rather than scientific reasoning. In bachelor quantum
mechanics, we struggle all the time in the foothills of a mountain of math. To see what really matters
(and what not), we need to climb that mountain.
The root of the problem with quantum mechanics lies here: we talk about reality and of properties of
objects most of the time in a naı̈ve way and most of the time this is a very reasonable approach. Significant
part of the history of philosophy, however, has been obsessed with the question how to logically capture
a concept of “reality” and where to place into that reality what we call “I” (or “we” for that matter) and
just how “real” it is that we see and seem and how such a possible (if possible) real world would make
its way into consciousness.
Physics does not answer these questions. However, in the form of quantum mechanics, it has treaded
upon their beginnings in a more striking and inescapable way than ever since the rise of modern science
with Galileo and Newton. If there is any belief in physics, then it is that what we know about the world
is not what we can imagine, but what we can reproducibly observe. What we call reality, in physics, is
a plausible abstraction from these facts, usually cast into the shape of a logical system, a mathematical
theory.
In classical mechanics, key elements of reality in that sense - velocity (later more precisely: momentum)
and position - appear to have been abstracted incorrectly from observation. In the system of classical
mechanics they both are considered as independently connected to observation. This was always known to
be an idea rather than a thoroughly studied fact. With increasing accuracy and diversity of measurement,
we became aware that there is no foundation in experiments for such an assumption. Even worse, by the
experimental violation of Bell’s inequalities, we know that the assumption is wrong.
The first few lectures of this course report a construction of classical and quantum mechanics based
on C ∗ algebra that employs many fewer ingredients and assumptions than the usual approach to either
theory. We cannot claim that there is all facts and no beliefs in this construction. However, this is true
for classical as well as quantum mechanics. We can claim, though, that quantum mechanics results, if
we omit from classical mechanics one belief that is not at all founded in observation: that we could, in
principle, do all possible observations on one and the same preparation of a system.
The use of Hilbert space and operators for doing quantum mechanical computations follow from that
very elementary difference to classical mechanics as a convenient representation of the underlying reality,
considering reality as that “plausible abstraction from known facts”.
Once we have reached that point, we will have gone through much of what is needed to deal with
bounded operators and we will have a rather complete mathematical foundation for many formal manipulations that are commonly used in quantum mechanics.
A decisive next step is to proceed to unbounded operators that represent abstracted, but central
objects such as position, momentum, or energy. This is rather heavy functional analysis of some mathematical beauty and great practical value.
Finally, we deal with scattering theory, which is an important topic but neglected in basic quantum
mechanics for its technical difficulty. On the one hand, it is esthetically satisfactory to lay a foundation
for the many formal manipulations of scattering theory with its mysterious “scattering states” and eigenfunctions of the continuum. In personal experience, I found such knowledge also of immediate practical
use for analytical work, and even for computational realization.
7
The C ∗ -algebra approach to classical and quantum physics
1
1.1
An analogy
State
Property
Measur.val. a
Probab. for a
After result a
Dynamics(I)
Dynamics(II)
Quantum
Classical
Vector Ψ from H
Prob. distr. ρ(x, p) on phase space
“Observable”, lin. op. A on H Function A(x, p) on phase space
Eigenvalues
of A
Function
values of A(x, p)
R
P
2
|ha|Ψi|
dx
dp ρ(x, p)
|ai∈Ker(A−a)
Ker(A−a)
Vector Φa ∈ Ker(A − a)
Prob. distr. φa (x, p) on Ker(A − a)
d
d
i dt Ψ(t) = HΨ(t)
ρ = LH ρ (Lie-derivative)
dt
R
Ψ(t) = Ut Ψ(0), hΨ(t)|Ψ(t)i ≡ 1 ρ(t) = Φt [ρ(0)], dxdp ρ(t) ≡ 1
(I) and (II) are equivalent alternatives
The symbol Φt designates a probability conserving, differentiable map, a flow on phase space. The
particular Lie-derivative LH may be more familiar as the Poisson bracket.
The formal analogy above is substantially due to an underlying structure that is shared by classical
b are part of a C ∗ algebra and states are positive bounded linear
and quantum theory: observables A
functionals on that algebra. The only difference between the two theories is that the algebra for classical
mechanics is commutative, but not in quantum mechanics. From this point of view, the Hilbert space is
only a convenient tool to represent (in the sense of representation theory) that structure, in case that the
algebra is non-commuting.
The following discussion relies on:
F. Strocchi, Mathematical Structure of Quantum Mechanics
W. Thirring, Mathematical Physics III: Quantum Mechanics
J.v. Neumann: Mathematische Grundlagen der Quantenmechanik
1.2
C ∗ -algebra
A C ∗ -algebra abstracts the essential algebraic and topological properties of matrices and bounded operators on a Hilbert space:
Definition: Algebra
An algebra over the complex numbers C:
1. A is a vector space
2. there is an associative multiplication: (P, Q) → O =: PQ ∈ A with the properties
P(Q1 + Q2 ) = PQ1 + PQ2 for P, Q1 , Q2 ∈ A
O(PQ) = (OP)Q
P(αQ) = α(PQ) for α ∈ C
∃1 ∈ A : 1Q = Q1 = Q
Definition: ∗-Algebra
An algebra is a called a ∗-algebra if there is a map ∗ : A → A with the properties (PQ)∗ = Q∗ P∗ ,
(P + Q)∗ = P∗ + Q∗ , (αQ) = α∗ Q∗ , Q∗∗ = Q. The element Q∗ is called the adjoint of Q.
8
Obviously this abstracts what you have encountered as “hermitian conjugate” of a matrix or an operator.
Definition: C ∗ -Algebra
An *-algebra A is called C ∗ if it has the following properties:
1. There is a norm 0 ≤ ||Q|| ∈ R with the usual properties (positivity, compatible with scalar multiplication, and triangular inequality) of a norm on a vector space and in addition
||PQ|| ≤ ||P|| ||Q||
||Q∗ || = ||Q||
||QQ∗ || = ||Q|| ||Q∗ ||
||1|| = 1.
2. The algebra is complete w.r.t. to that norm (Banach space).
We will discuss these properties later in more detail.
1.3
Classical mechanics
Before discussing this in more abstract terms, let us note that actually the observables of classical mechanics are part of a C ∗ algebra, at least if we restrict ourselves to what can be really measured. Under
the same restriction, states of classical systems are (positive) linear functionals on that algebra.
1.3.1
Observables
The idea of an “observable” is that is represents an actual measurement. In that sense, a classical
observable is a function on phase space that can be realized in one specific experimental setup. Its values
A(~r, p~) is the number extracted from the measurement, given the system is at the phase-space point
~r, p~ ∈ Γ, e.g. Γ = R2n The requirement of realization in an experiment restricts observables in this sense
to bounded functions:
||A|| := sup |A(~r, p~)| < ∞,
(1)
Γ
which also defines our norm. We see that this operational definition excludes much of what we would
consider “properties” of a system (position, momentum, energy, etc.), but it includes everything we
could ever measure in an experiment, which is the more scientific statement. We will re-encounter
this fundamental problem of abstraction, when we will deal with the unbounded operators of quantum
mechanics.
The rest is easy and it will be left as a first exercise to construct the C ∗ algebra that contains the
classical observables.
The abstract “properties” of a system can be considered as limits of the observables. These limits
may exist, but they will not in general be themselves observables, because the series is not convergent in
the algebra norm. If a sequence of measurement arrangement is convergent, then, by the completeness
of the C ∗ algebra, the limit is also an observable.
1.3.2
States
As an idealization, the state of a classical systems is fully defined by a point in Γ. However, this is not
what any scientist ever has been able to observe. If we restrict ourselves to what we can really do, we
arrive at a slightly different “operational” definition of a state: It is a procedure that we can repeat many
times with what we consider high precision. That “preparation procedure” defines what we call a state.
9
Repeated observations of the same observable using the same preparation procedure will give a certain
distribution of observed values. We can compute the expectation value of the observables for a given
preparation procedure ω as the average over many measurement results ai :
hAiω =
N
1 X
ai ∈ R
N i=1
(2)
The individual value of one measurement is of little significance because of the statistical uncertainty of
our preparation procedure.
As a minimal requirement for “high precision” (for the purpose of a given experiment) is that the
measurement values scatter little, for example that
hAn i ≈ hAin
(3)
In that sense, a state is a functional on the observables, more generally on the C ∗ -algebra that contains
the observables: it maps any element A ∈ A into hAiω ∈ C, in physics terms: the state (=preparation)
of the system determines the average value of a measurement (and also its variance etc.). We usually
construct observables as to be real numbers.
In our specific example we see three properties that we will consider the defining properties of a state,
linearity, normalization h1iω = 1, ∀ω, and positivity:
Definition: State of a system
A state is a normalized positive linear functional on the the C ∗ algebra of bounded functions.
1.4
Justification of the C ∗ algebraic nature of observables
Above we have seen that classical mechanics is “embedded” in a C ∗ algebra, i.e. any observable that
corresponds to realizable experiments can be considered as an element in a larger structure, which forms a
C ∗ algebra. The assumptions underlying this are very “natural”, in the sense that this is how we perceive
our experimental procedures. However, one must warn already here, the C ∗ structure cannot be fully
motivated by such considerations: it may well be that there is a meaningful (maybe even the ultimately
correct?) description of nature that cannot be embedded into a C ∗ algebra but still fulfills all procedural
requirements (linearity, normalizability, etc.) that we put a priori into a scientific theory. The Hilbert
space formulation follows from the C ∗ -structure and the definition of states. It is a convenient technical
representation of the algebra and it is guaranteed to have the essential properties of meaningful theory.
There is further structure, though, not required by reason a priori. Therefor this specific form may not
be the only choice. However, we find to this point no statements in the theory that would contradict
experiment.
We will now justify a few aspects of the algebraic form of the physical theory and than make the leap
of faith that it is best to realize these properties as part of a C ∗ algebra. Much of the justification was
already anticipated in the discussion of the classical mechanics case.
1.4.1
Observables
An observable A is an experimental apparatus that produces real numbers a as its results. Clearly, we
can, e.g. just by changing measurement units, multiply an observable by a real number λ. We can form
powers of the measurement results by forming the power of the result a. We can define an observable A
as positive if any measurement results in a non-negative number a.
1.4.2
State
A state ω of a physical system is a procedure of repeated experiments that yields a (real) expectation
value ω(A) for any observable. In the sense of empirical science, a state is completely characterized, if we
10
know the expectation values of all possible observables. Two states that give the same result for anything
we measure are plausibly called equal. By its definition as the average over repeated measurements, we
find the properties for a given observable A:
ω(λA) = λω(A),
1.4.3
ω(Am + An ) = ω(Am ) + ω(An )
(4)
Normalization and positivity of a state
The properties
ω(1) = 1 and ω(A) = ω(B 2 ) ≥ 0
(5)
follow easily.
1.4.4
Equality of observables
We can reason in a similar way for the observables: any two observables, that give the same result for all
possible states defined are equal:
ω(A) = ω(B) ∀ω ↔ A = B.
(6)
Here we assume that a state can be applied to all observables. This is not completely obvious if we
adhere to a strict operational description as a certain way of preparing a systems may be incompatible
with certain ways of measuring things. Of course, as soon as we abstract the procedure into some internal
property of the “system” we should be able to perform any possible measurement on it.
1.4.5
C ∗ -structure
Linear structure and norm can be plausibly deduced along the same lines (see Strocchi): for the linear
structure, we use that we call two observables equal, when their expectation value for all states ω is equal.
This, in particular, means that we know an observable, if we know its expectation value for all states
A + B = C : ω(C) = ω(A + B) ∀ω
(7)
We can define the norm of an observable as “the largest possible expectation value in any experiment”.
As it is not a priori obvious whether this largest value will ever be actually assumed by for a single state,
we use the supremum sup rather than the maximum:
||A|| = sup ω(A).
(8)
ω
Problem 1.1: S how that this has the properties of a norm in a linear space.
One may ask, however whether the observable C = A + B can be mapped onto a single realizable
experiment. If not, it may define an object that does not comply with the original definition of observables.
Our forming of powers of C relies on this operational prescription: we need to take powers of the individual
measurement results ai + bi . Now, if we cannot obtain ai and bi for the same individual measurement i,
we have no procedure to form a power of C by averaging over powers of ci = ai + bi .
At this point we withdraw from the real world of experimental arrangements into mathematics by
assuming that, for whichever are the mathematical objects that represent an observable C, the powers of
C can be formed in a meaningful way. This is certainly the case in classical and quantum mechanics, but
we see no reasons for considering it an a priori necessity of a theory describing experimental practice.
If we accept this, we can make a few more steps to prove the triangular inequality for the norm, but
this is where our beautiful story ends. One can proceed to define symmetric products using (A + B)2 as
(A + B)2 − A2 − B2 = AB + BA =: A ◦ B
which results in an algebraic structure that reproduces much of quantum mechanics.
11
(9)
In some, but not all cases, such an algebraic structure with a symmetric product can be realized
as a subset of a C ∗ algebra (embedded in the C ∗ -algebra). It is no obvious, whether cases where this
cannot be done may be physically interesting. We terminate our discussion here and assume that the
observables are embedded in a C ∗ -algebra. Our goal is a different one: we will show that the structure of
a C ∗ -algebra together with a state generates a Hilbert space and that, actually, any Hilbert space with
a set of (bounded) observables on it is a (possibly infinite) is equivalent to a sum of such Hilbert space
representations for several states ωi . This includes classical mechanics, which has commutative algebra
of observables.
1.4.6
Basic structure of a physical theory
1. A physical system is defined by a C ∗ -algebra that contains all its observables
2. A state is a positive normalized linear functional on the C ∗ algebra
3. An observable is fully defined by its expectation values on all possible states. Conversely, a state is
fully defined by its expectation values for all elements of the algebra.
Note:
• There is a duality between states and observables, as we know it from linear functionals on vector
spaces.
• In reality, we know exactly two theories that comply with this definition: classical and quantum
mechanics. In case of classical mechanics, it appears to be an overblown definition, but it precisely
fits quantum mechanics as we know it.
12
C ∗ algebra
2
Definition: Algebra
An algebra over the complex numbers C:
1. A is a vector space
2. there is an associative multiplication: (P, Q) → O =: PQ ∈ A with the properties
P(Q1 + Q2 ) = PQ1 + PQ2 for P, Q1 , Q2 ∈ A
O(PQ) = (OP)Q
P(αQ) = α(PQ) for α ∈ C
∃1 ∈ A : 1Q = Q1 = Q
Definition: ∗-Algebra
An algebra is a called a ∗-algebra if there is a map ∗ : A → A with the properties (PQ)∗ = Q∗ P∗ ,
(P + Q)∗ = P∗ + Q∗ , (αQ) = α∗ Q∗ , Q∗∗ = Q. The element Q∗ is called the adjoint of Q.
Obviously this abstracts what you have encountered as “hermitian conjugate” of a matrix or an operator.
Definition: C ∗ -Algebra
A *-algebra A is called C ∗ if it has the following properties:
1. There is a norm 0 ≤ ||Q|| ∈ R with the usual properties (positivity, compatible with scalar multiplication, and triangular inequality) of a norm on a vector space and in addition
||PQ|| ≤ ||P|| ||Q||
||Q∗ || = ||Q||
||QQ∗ || = ||Q|| ||Q∗ ||
||1|| = 1.
2. The algebra is complete w.r.t. to that norm (Banach space).
2.1
Banach space
Definition: Linear space
A linear space (≡ vector space) over the complex numbers C is a set where an addition and multiplication
by scalars is defined.
Examples (we will use these to illustrate the distinctive properties of C ∗ -algebras)
1. Vectors in Cn
2. Complex n × n matrices
13
3. Polynomials of complex arguments
4. C r - the set of r-times continuously differentiable functions
5. Analytic functions
Definition: A normed space
is a vector space V where a norm || · || : V → R is defined with the properties
1. ||~v || ≥ 0
2. ||α~v || = |α| ||~v ||
3. ||~u + ~u|| ≤ ||~u|| + ||~v || (triangular inequality)
4. ||~v || = 0 ⇔ ~v = 0
Examples of norms
For the vector spaces we can define norms in various ways
1. Cn , the p-norms
"
||~v ||p =
n
X
#1/p
|vi |
p
,1 ≤ p ≤ ∞
i=1
As a limiting case p = ∞:
||v||∞ = max |vi |
i
c, we also can define (entry-wise) p-norms
2. On the n × n-matrices M

1/p
n
X
c||p = 
||M
|mij |p 
i,j=1
An interesting case is p = 2 (Hilbert-Schmidt norm), where we can write

1/2
n
h
i1/2
X
c||2 = 
cM
c∗ )
||M
|mij |2 
= Tr(M
i,j=1
(We denote by ∗ the adjoint = hermitian conjugate). If we were to arrange the entries of the matrix
c|| = ||~vM ||2 .
in a vector, say ~vM , this becomes ||M
c, we may define the norm
3. On the n × n-matrices M
c|| = sup ||M~
cv ||,
||M
||~
v ||=1
cv || we use the 2-norm. For linear operators, this is the “operator
where for the vectors ||v|| and ||M~
norm”, in case of matrices the largest magnitude of the eigenvalues.
As the space is finite, we could just as well have written a max instead of sup. For finite matrices,
although this matrix norm and all entry-wise p-norms are distinct, they are equivalent in their
topological consequences: any open set with respect to one norm contains an open set w.r.t. to the
other norm. The analogous constructions in infinite Hilbert spaces generate profoundly distinct
limiting behavior and the choice of a “suitable” norm will be essential.
14
4. Polynomials P (x) on a compact set, supremum norm.
5. r-times continuously differentiable functions, supremum norm.
6. If a measure for integration is defined: generalization of the p-norms.
Definition: Completeness
A space is called complete if any Cauchy-sequence has its limit in the space. A normed space that is
complete w.r.t. the norm is called Banach space.
2.1.1
Comment
Here we begin to differentiate a bit more: of the examples above, the finite-dimensional spaces are complete (they always are), further the continuous functions on a compact set (r = 0) and the functions with
the p-norm. Polynomials do not always converge to polynomials, similarly one easily looses differentiability, r > 0, in a limiting process.
2.2
Classes of elements of a C ∗ algebra
1. Hermitian: Q∗ = Q
2. Unitary: Q∗ Q = QQ∗ = 1
3. Normal: Q∗ Q = QQ∗
4. Projector: Q2 = Q = Q∗ (more strictly: orthogonal projector)
5. Positive: ∃C ∈ A : Q = C ∗ C
2.3
Open set
If a norm is given on some set X, we can define an open ball of radius r > 0 around z0 : Br (x) = {z0 ∈
X|||z0 − y|| < r. We can then define:
Definition: Open set
is U ⊂ X, where for any u ∈ U there exists B (u) ⊂ U .
We see that the norm allows to construct to define a system of open sets for X, one says the norm
“induces a topology”.
2.4
The spectrum of A ∈ A
Definition: Resolvent set and spectrum
The set
z ∈ C : ∃(A − z)−1
is called resolvent set of A. Its complement σ(A) is called the Spectrum of A.
15
(10)
2.4.1
Series expansion of the resolvent
(z − A)−1 = z −1
∞
X
(z −1 A)n .
(11)
n=0
That series, depending on the size of |z|, may or may not be convergent.
2.4.2
The resolvent set is an open set
Let z0 be in the resolvent set and let |z − z0 | < < 1/||(z0 − A)−1 ||. Then the formal series
(z − A)−1 = (z − z0 + z0 − A)−1 = (z0 − A)−1
∞
X
n
(z − z0 )(z0 − A)]−1
(12)
n=0
is norm-convergent and therefore has its limit in the C ∗ algebra. To see norm-convergence, we need to
the estimate
||
N
X
[(z − z0 )(z0 − A)]−1 ||
n=0
≤
|{z}
triangular
≤
|{z}
||PQ||≤||P||||Q||
N
X
||[(z − z0 )(z0 − A)]−n ||
(13)
|(z − z0 )|n ||(z0 − A)]−1 ||n
(14)
n=0
N
X
n=0
and choose |z − z0 |||(z0 − A)−1 || < 1 or z ∈ B (z0 ) with = 1/||(z0 − A)−1 ||
2.4.3
Algebra element properties and spectrum
There is a close relation between the spectrum and the class of an element and its spectrum. In fact, the
spectral properties fully characterize hermitian, unitary, positive and projector elements, if we assume
that Q is normal. You will study this as an exercise.
2.5
Problems
The first few of the following problems are drawn from the “Preliminaries” section of Reed-Simon, where
you also can find much of the solutions. It may be (I hope) that you are familiar with some of the math,
in that case consider the exercises as reminders.
Problem 2.2: Normed vector space Let C[0,1] denote the continuous real-valued functions
on [0, 1].
(a) Convince yourself that with pointwise addition this is a linear space.
(b) Show that the functionals
Z
||f ||∞ = max |f (x)| and ||f ||1 =
x∈[0,1]
1
dx|f (x)|
(15)
0
are norms.
(c) Generalize the norm and proof of the norm properties for the space continuous functions A(~r, p~)
over a compact subset of phase-space Γ ⊂ R6 .
16
Problem 2.3: Convergence, Cauchy sequence Remember the definition of convergence:
a sequence of elements {xn , n = 0, 1, . . . , ∞} from a normed (vector-) space V is said to converge to an
element
x ∈ V if ||x − xn || → 0 as n → ∞. Two other important concepts are open ball B (x) =
{y ||y − x|| < } and neighborhood of x, Nx : ∃ and B (x) ⊂ Nx , i.e. a set that contains x and also
an open ball around x.
(a) Prove that any convergent sequence is a Cauchy sequence.
(b) xn → x if and only if for each neighborhood Nx of x there exists an M such that m ≥ M implies
xm ∈ Nx .
Problem 2.4: Operator norm Let V denote a vector space over the complex numbers C with
a norm || · ||. Let T denote a linear transformation V → V
T : T v = u,
(16)
(a) Show that linear transformations form themselves a linear space B, if one defines addition of two
maps by
T, S ∈ B : (T + S)v = T u + Su
(17)
and in similar way multiplication by a scalar.
(b) Assume now that V is finite-dimensional (i.e. there is a finite basis) Show that
||T || := max ||T v||/||v||
v∈V
(18)
defines a norm on B (||T v|| is the norm on V).
(c) Consider the case where V is infinite-dimensional. Show that
||T || := sup ||T v||/||v||
(19)
v∈V
can diverge, i.e. not every linear operator is bounded. Give the simplest possible examples for
this with V = L2 (dx, R). On l2 (= “infinitely long vectors” (a0 , a1 , ...an , ...), can you give an
infinite-dimensional matrix that is unbounded?
Continuity is an essential property of maps, as it allows us to take limits, i.e. to approximate, e.g. an
idealized measurement by a sequence of increasingly accurate measurements and assume that the perfect
measurement would be close to our “increasingly accurate” measurements. In a sense, it is the idea of
“continuity” makes the concept of “accuracy” meaningful.
For functions f between sets with a norm, continuity can be best understood as “the limit of the
function values is the function value of the limit”, i.e.
f is continuous ⇔ f (xn ) → f (x) if xn → x.
(20)
This is not the most general definition of continuity, but it serves all our purposes and matches intuition.
A bounded operator is an operator with finite operator norm (problem above). In linear spaces,
boundedness of a linear map and continuity are intimately related.
Problem 2.5: Boundedness and continuity Let E and F be two normed linear spaces and
b : E → F be a linear map. Show that the following statements are equivalent:
let B
b is continuous at the point 0 = x ∈ E
1. B
17
b is bounded
2. B
b is continuous everywhere
3. B
w.r.t. to the supremum-norm.
Hint: This is a standard result for linear maps. For those who have not been exposed to it, here a few
hints: show 1 ⇒ 2 ⇒ 3 ⇒ 1. In the first step use the fact that in a linear space, any vector ||x|| = 1 can
b
be scaled to length . Apply this to ||Bx||/||x||
that appear in the definition of the norm. For the step
bounded ⇒ continuous, observe that ||B(xn − x)|| < ||B||||xn − n|| by construction of the operator norm.
Problem 2.6: Continuity of maps, completeness Let E and F be two Banach spaces and
let B(E, F) be the space of the continuous linear maps from E to F.
(a) From the previous exercises we know that the linear maps themselves form a linear space. Show
b B := supx∈E ||Bx||
b F /||x||E , B
b ∈ B is a norm for B. Here || · ||E , || · ||F denote the norms in
that ||B||
the respective spaces.
(b) Show that B with || · ||B is a Banach space.
bn } defines a continuous
Hint: For completeness of B one needs to show that a Cauchy sequence {B
b
b
linear map B := limn→∞ Bn . Using completeness of F, one sees that it definitely defines a map
into F. Boundedness can be shown with a little effort, which then also implies continuity by the
previous problem.
Problem 2.7: Positivity Show that for hermitian A the element ||A||1 ± A is positive .
Hint: Write ||A|| ± A = C 2√for hermitian C and explicitly construct C using the fact that there is a
convergent Taylor series for 1 ± x. Use the properties of the algebra to show the series converges with
a limit in the algebra.
Problem 2.8: Positivity and boundedness Show that any positive linear functional on a
C ∗ algebra is bounded.
Hint: Use Cauchy-Schwartz and positivity of ||A|| ± A for hermitian A.
Problem 2.9: Resolvent Let A ∈ A be an element of a C ∗ algebra. Then also A − z ∈ A, where
we use the notation A − z for A − z1, z ∈ C. We call (A − z)−1 : (A − z)−1 (A − z) = (A − z)(A − z)−1 = 1
the resolvent RA (z) of A at z if it exists and (A − z)−1 ∈ A.
(a) Show that the formal series expansion
(z − A)−1 = z −1
∞
X
(A z −1 )n
(21)
n
converges for |z| > ||A||.
Hint: Use the properties of the norm on a C ∗ -algebra to show the the series of finite sums is
Cauchy.
(b) Conclude from the above that (A − z)−1 ∈ A for ||A|| < |z|.
Problem 2.10: Classes of algebra elements and spectrum Normality is an important
property that is not at all guaranteed.
(a) Using the C ∗ algebra of n × n matrices, give an example (the example ;-)) of a non-normal matrix.
18
Properties like hermiticity, projector, unitarity, positivity of elements of a C ∗ algebra are reflected in the
spectra of the respective elements. This is trivial once we can use the spectral theorem, but it can be
proven directly as well. For the following cases this can be done without any particular tricks:
(b) σ(A∗ ) = σ(A)∗
Hint: Of course, you should prove this for the complement of the spectrum, the resolvent set, first.
(c) σ(U ) ⊂ (unit circle) for U unitary.
Hint: Determine ||U || and ||U −1 || and Taylor-expand (U − z)−1 and (U ∗ − z∗)−1 = (U −1 − z ∗ ).
Problem 2.11: Trace For operators in infinite dimensional Hilbert space, the trace formally defined
as in finite dimensions
Tr A =
∞
X
hi|A|ii,
(22)
i=1
for any orthonormal (ON) basis {|ii}. Verify formally the following properties:
(a) The definition is independent of the choice of the basis {|ii}.
(b) Tr AB = Tr BA and Tr ABC = Tr BCA.
(c) Does Tr ABC = Tr BAC hold?
(d) For normal operators in finite dimensions det[exp(A)] = exp Tr A.
New issues arise in infinite dimension:
(e) Is the trace guaranteed to exist? At which places does one need to treat the fact that the formal
sums are infinite-dimensional?
(f ) Suppose | Tr A| < ∞ and | Tr B| < ∞, does | Tr AB| < ∞ hold?
(g) Suppose | Tr A| < ∞ but | Tr B| = ∞, can Tr AB exist? (Try to find an example).
19
2.6
State on a C ∗ algebra
A linear functional ω on a C ∗ algebra A is called positive if ω(A∗ A) ≥ 0∀A ∈ A. If ω(1) = 1, we call it
a state.
Examples
1. In quantum mechanics: hx|Axi
R
2. In classical mechanics: Γ dxdpρ(x, p)A(x, p) for ρ ≥ 0.
3. On matrices, Tr ρA for a “density matrix” ρ, i.e. a positive matrix (equivalent hermitian with
non-negative eigenvalues) with Tr ρ = 1.
4. Positive measures on function spaces
2.6.1
Notes
1. There holds a Cauchy-Schwartz inequality, i.e.
|ω(A∗ B)|2 ≤ ω(A∗ A)ω(B ∗ B)
(23)
This will be left to you as an exercise.
2. Positivity of ω implies its boundedness.
3. As always with linear maps, boundedness implies continuity. Intuitively, if two vectors do not differ
too much, ||P − Q|| < , the also the images do not differ too much: ω(P − Q) ≤ Cω ||P − Q||, which
is the motivation for general definition of continuity: the pre-image of any open set is an open set.
(See exercises).
Problem 2.12: Cauchy-Schwartz Prove the generalized Cauchy-Schwartz inequality for states
ω
|ω(A∗ B)|2 ≤ ω(A∗ A)ω(B ∗ B)
(24)
Hint: This can be modeled after one standard form of proving Cauchy-Schwartz. Follow these steps:
(a) study 0 ≤ ω((A∗ + B ∗ )(A + B)), (b) show that ω(A∗ B) can be considered as real without loss of
generality; (c) use positivity of ω to show that ω(A∗ B) = ω(B ∗ A), (d) look at the square of (a).
2.6.2
The states form a convex set
ω = αω1 + (1 − α)ω2 ,
α ∈ [0, 1],
ω1 , ω2 states
(25)
is a state. They do not form a linear space, e.g. 2ω is not a state, if ω is one.
2.6.3
Pure and mixed states
There are special states, the pure states, which cannot be written as a linear combination of states.
Conversely, one can prove that any state can be written as a convex combination of pure states:
X
X
ω=
αi ~ui ,
αi = 1, νi pure
(26)
i
i
States that are not pure are called mixed.
20
We know and we will provide the prove later that a quantum mechanical wave function ψ defines a
pure state ωψ
b = hψ|A|ψi.
b
ωψ (A)
We see that this, according to our general principles, is not the only possibility for a state. The more
general mixed states come by the name of density matrix that is
X
X
ωρ =
ρi ωψi ,
ρi = 1,
(27)
i
i
X
b =
ρi ωψi (A)
with the action
b =
ωρ (A)
i
X
i
21
b ii
hψi |A|ψ
(28)
3
Gelfand isomorphism
The essence of this isomorphism is that Abelian C ∗ algebras are isomorphic to algebras of functions and
further that the algebra spanned by {1, A, A∗ } for normal A is equivalent to the functions over σ(A).
This contains the essence of the spectral theorem for normal bounded operators.
Definition: Abelian algebra
is an algebra where the product is commutative.
In general, in mathematics, the term “homomorphism” designates a map from algebra into another
that preserves the algebraic operations in the sense that the image of a sum is the sum of the images,
the image of a product is the product of the images. For *-algebras, one defines the
Definition: Algebraic *-Homomorphism
A map π : A → B between *-algebras A, B is called *-homomorphism if
1. π(AB) = π(A)π(B)
2. π(αA + βB) = απ(A) + βπ(B)
3. π(A∗ ) = π(A)∗ .
Definition: Character
An algebraic *-homomorphism χ of an Abelian C ∗ algebra into the complex numbers C is called character. We denote the set of all characters of an algebra A as X(A).
Illustration
1. Let ~e be a joint eigenvector of all elements the algebra formed by commutating matrices. Then
b = ~e · A~
be is a character of the algebra.
χ~e (A)
2. Bounded functions on a compact set: values at a given element from the compact set.
Comments
1. (χ(A) − z)−1 exists if z 6∈ σ(A), therefore σ(χ(A)) ⊂ σ(A).
2. A character is trivially positive χ(A∗ A) = χ(A)∗ χ(A) ≥ 0. Therefore it is also bounded.
3. Any character is a state: it is a positive, linear map A → C and χ(A1) = χ(1)χ(A) shows
normalization.
4. Just as the states do not form a linear space (why?), the characters do not form a linear space.
5. As states, also characters are bounded by ||A||: |χ(A)| ≤ ||A||.
6. The Cauchy-Schwartz inequality that holds for states and therefore for characters |χ(A∗ B)|2 ≤
χ(A∗ A)χ(B ∗ B) implies |χ(A)| ≤ ||A||: choose B = 1 and use that 0 ≤ χ(||A||2 − A∗ A) = ||A||2 −
χ(A)χ(A)∗ .
7. Characters are pure states: let χ be a character and ω1 and ω2 two general states, then (α1 ω1 +α2 ω2 )
(α1 + α2 = 1) will not be a homomorphism
22
8. There exists a pure state such that χ(A) = ||A||: this conforms with our analogy that characters
b is the modulus of its largest eigenvalue.
are correspond to eigenvectors and norm of a matrix ||A||
One can explicitly construct that state as follows: Given any A ∈ A, we construct the subspace
of all vectors α1 + βA∗ A. On that subspace, χs (α1 + βA∗ A) = α + βA∗ A is a positive functional
with χs (1) = 1 and χs (A∗ A) = ||A||2 . This χs is only defined on the subspace. However (theorems
of Hahn-Banach and Krein) there are states on all of A that behave as χs on the subspace. Now
that we know that such states exist, let us denote the states with ω(A∗ A) = ||A||2 by Z: Z is a
clearly a convex set. The extremal states χe ∈ Z are pure states: because of ω(A∗ A) ≤ ||A||2 ,
χs = αω1 + (1 − α)ω2 implies ωi (A∗ A) = ||A||2 , i.e. ωi ∈ Z and χe would not be extremal in Z.
We had not proven that states are bounded. For characters that prove is really easy and all that we
need for now: For hermitian A we know 0 ≤ ||A|| ± A, therefore 0 < ||A|| ± χ(A), i.e. |χ(A)| ≤ ||A||. For
general A, we note that A∗ A is hermitian and therefore (using properties of || · ||
||A||2 = ||A∗ A|| ≥ χ(A∗ A) = χ(A)∗ χ(A) = |χ(A)|2 .
3.1
(29)
Weak *-topology
Strictly a topology on any set is given, if we define all open subsets of the set.
Topology is required to define continuity: a function is continuous at a point, if the pre-image (“Urbild”) of any open neighborhood of a image of the point is an open neighborhood. This is an abstraction
and generalization of our intuition for continuous functions on Cn .
We will want to define continuous functions from the character set X(A) into the real numbers. We
need a topology, i.e. we need to define the “open sets” ⊂ X(A). Now the characters are subsets of the
linear functionals on the algebra: they are subsets of the dual space of the algebra.
The open sets in the dual space are defined at those where any point space W ∗ in the set contains a
set Bv, (W ∗ ) = {U ∗ ∈ V||U ∗ (v) − W ∗ (v)| < } for all v ∈ V .
We define when we call a sequence in the dual space as convergent in the sense of this topology: a
sequence of elements χn from the dual space of some vector space V is said to converge in the sense of
weak convergence, if χn (v) converges as a sequence in C for each v ∈ V.
Illustration and comments on weak convergence In finite dimensions most topologies are equivalent and also the distinction between the a linear space and the linear functionals on that space is not
essential. In infinite dimensions the distinctions become crucial.
1. Continuous functions on a compact set S: a character χx maps a function f ∈ C0 : χ(f ) = f (x)
into its value at one point. A sequence of characters χxn converges, if the function values f (xn )
converge at that point, which, as f is continuous, is the case if xn converge in the sense of the
topology of S. (You may recognize the δ-functions as characters of that algebra.)
2. n-component complex vectors ∈ Cn : we do not have an algebraic structure and therefor no characn
∗
ters. We havePa dual, of course.
P Choosing a basis ei ∈ C , any linear functional W can be written
∗
∗
as W (v) = i W (ei )vi = i wi vi , where the wi are components in the dual space. A sequence
(n)
Wn∗ converges, if each of its components wi converges. This dual is the isomorphic to (“for all
practical purposes the same as”) the original space.
3. On a finite dimensional vector space we can use an alternative concept with equal result: As we
have a norm for the v ∈ Cn and as we can identify every W ∗ ∈ dual(Cn ) with some w ∈ C2 , we
can use that norm also for the W ∗ : ||W ∗ || := ||w||. Sequences that are convergent w.r.t. the norm
(“strongly convergent”) are also weakly convergent. In finite dimensions, also the converse is true.
4. In infinity dimensions weak convergence does no longer imply strong convergence. Hilbert space:
weak convergence does not imply strong convergence: trivial, but also typical example is ~vn : (~vn )i =
δin
23
5. Example from physics: a (purely continuous) spreading wave packet converges → 0 in the weak
topology!
A ∈ A is a continuous function on X(A): We are seemingly going in circles: we just introduced the
characters χ as functionals on A. We can turn the roles around and consider the elements of the algebra
as functions on the set of characters
A 3 A : X → C : χ → χ(A).
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As a direct consequence of the construction of the weak-* topology, A is a continuous function w.r.t. the
*-topology on X(A).
Problem 3.13: Continuity of A : X(A) → C Show that the map A : χ → χ(A) is continuous
w.r.t. the weak *-topology on X.
3.2
Gelfand isomorphism
An Abelian C ∗ algebra is isomorphic to the continuous functions C(X) on the characters set X = X(A)
if we use the weak *-topology for X and the norm C(X) 3 f : ||f || = supχ |f (χ)|.
Isomorphic By isomorphic we denote maps that are bijections and that conserve the essential structural
characteristics such as algebraic properties, *-map, and norm. In colloquial language, things that are
isomorphic “are the same”, just represented in different ways.
Outline of the proof Even if this remains very incomplete, the reasoning with approximations by
algebras, dense sets, and compact sets if of general value and therefore useful to look at:
1. We have seen that any element A of the algebra defines a continuous function fA ∈ C(X). Let
fA , A ∈ A denote the set of these functions. Clearly, as characters are *-homomorphisms, any
algebraic combination of these fA ’s matches the algebraic combination of the corresponding A’s
e.g.
fAB (χ) = χ(AB) = χ(A)χ(B) = fA (χ)fB (χ),
(31)
and similarly for all other algebraic operations. That means that each element of the algebra
re-appears as a function on X(A) and that these functions form a *-algebra.
2. Next we convince ourselves that the C ∗ -algebra norm maps onto the (supremum) norm of fA ∈
C(X):
!
||A|| = sup |fA (χ)| = sup |χ(A)|.
χ
(32)
χ
As also all χ are states, |χ(A)| ≤ ||A||. Here we use the fact that all characters are states and,
without proof, that the pure state with |ω(A)| = ||A|| is a character, i.e. the supremum over all χ
is indeed the norm.
3. We have shown in an exercise that fA is a continuous function, i.e. the map A → fA is injective
into C(X(A)).
4. The main mathematical task is to show that A → fA is also surjective, i.e. that indeed all continuous functions f ∈ C(X) have a correspondence in the algebra. Rather then proving anything
here, we appeal to your knowledge of a fact from the theory of ordinary functions (Weierstraß):
polynomials approximate any continuous function on a compact set arbitrarily well in the sense of
the supremum norm: they are dense in the continuous functions. Compactness must be understood
for the *-topology on X.
24
In fact, this holds not only for polynomials but for any algebra of complex-valued functions on a
compact set (not proven here). Now our fA definitely form an algebra of complex valued functions
on X(A), which is compact. Therefore the fA are dense in C(X). This means that any function
f ∈ C(X) can be approximated by a sequence fAn with An ∈ A. As ||An || = ||fAn || we know that
An is a Cauchy sequence and An → A ∈ A with fA = f .
Comment on this sketch: There are a few very important holes in this; but as it is not our main
purpose to follow up on this, we only list these holes here clearly:
1. The proof that the pure state with ω(A) = ||A|| is a character is missing. This proof will be
delivered later: the GNS construction shows that pure states for abelian algebras indeed produce
*-homomorphisms into the complex numbers, i.e. characters.
2. We have not discussed compactness, which for the special case of weak *-topologies can get tricky.
Nonetheless we assert that the character set is compact as needed as input for Weierstraß.
3. Even if we take Weierstraß for granted, its generalization for algebras needs to be looked up in math
literature, e.g. [?]
3.2.1
Terms: Closure and dense sets
Closure of a subset Let D be a subset of a space X where a distance is defined between all elements
(metric space). Then the closure D of D is
D = D ∪ {x ∈ X|∃dn → x}
Dense subset
A subset D of a metric space X (space with a norm) is called dense in X if
D = X.
3.2.2
(33)
(34)
The spectral theorem in nuce
Let A be a normal element of any (also non-commutative) C ∗ -algebra A and consider the sub-algebra
generated by {1, A, A∗ }. Normality implies that the algebra of all polynomials of A and A∗ is Abelian.
If we consider the closure of the algebra of the polynomials, we obtain again a now Abelian C ∗ algebra.
(Strictly, one needs to show that commutativity survives the limit). By the Gel’fand theorem, this algebra
is isomorphic to the functions on the character set. This implies, in particular, that
∃(A − z)−1 ⇔ ∃(fA (χ) − z)−1 ∀χ
(35)
σ(A) = ran(fA )
(36)
or
i.e. the “range” of the function fA (=image fA (X)) is just the spectrum of A. Considering now that
fA (χ) = χ(A)
∈ σ(A)
(37)
we see that a character maps A into the spectrum. We can also consider polynomials P
fP (A) (χ) = χ(P (A)) = P (χ(A)) == P (χ(A)) = P (a) for a := χ(A) ∈ σ(A)
(38)
to see that all f ’s are just functions on the spectrum of A.
To make the spectral theorem as you know it complete, we need the Hilbert space. This rabbit we
will magically pull out using just an innocent state for out top-hat.
Problem 3.14: Spectra and Gelfand isomorphism Use Gelfand isomorphism to prove
the spectral characterization of hermitian, unitary, positive and projector elements of a C ∗ algebra.
25
4
Representations in Hilbert space
4.1
Operator norm
For one of the exercises we already introduced the boundedness of a linear map as the property that a
vector “does not grow too much” when the the map is applied:
||Av|| ≤ C||v||
∀v
(39)
for some C > 0. The smallest possible such number is the norm ||A||:
Definition: Norm of an operator
Let A : H → H be a linear map of a Hilbert space H onto itself. We designate as the norm of A on
Hilbert space H the number
||A|| = sup ||Aψ||,
(40)
||ψ||=1
where on the rhs. || · || denotes the vector norm of H. An bounded operator A is an operator with a
finite norm ||A|| < ∞.
Note: the operator norm has all the properties that we asked from the norm of a C ∗ -algebra.
4.2
Representation of an algebra
We have dealt with algebras reduced to their abstract properties, i.e. by defining the results of addition,
multiplication. In practice, such an algebra is realized by objects like matrices, differential operators etc.
One and the same algebra can be realized in different ways. Also, the realization may only represent part
of the complete properties (it may not be an isomorphism).
Definition: Representation
A representation π of a C ∗ -algebra A is a *-homomorphism from A into the bounded operators B(H) on
a Hilbert space H. If ker(π) = 0 ∈ A the representation is faithful. Two representations π1 on H1 and
π2 on H)2 are equivalent, if there is an isomorphism U : H1 → H2 such that
π2 (A) = U π1 (A)U −1
4.2.1
∀A ∈ A.
(41)
Note
• Continuity of π follows from positivity: π(A∗ A) = π(A)∗ π(A) is a positive operator on H. This
implies in particular
||π(A)∗ π(A)ψ|| ≤ ||π(A)||2 ||v||∀||v||
(42)
0 ≤ A∗ A ≤ ||A∗ A||1 ⇒ ||π(A)∗ π(A)|| ≤ ||(||A∗ A||)1|| = ||A∗ A||
(43)
0≤
(44)
• As representations may not be faithful, we can have ker(π) 6= 0. In that case
26
4.3
Irreducible representation
Definition: Invariant subspace
Let V ⊆ H be a subspace of H, i.e. V is a linear space. V is called an invariant subspace of H for a
representation π, if
~v ∈ V ⇒ π(A)~v ∈ V, ∀A ∈ A
(45)
Definition: Cyclic vector
A vector ~c ∈ H is called cyclic vector for a representation π, if
C := {π(A)~c|A ∈ A}
(46)
C = H.
(47)
is dense in H, i.e.
Definition: Irreducible representation
The following two properties are equivalent
1. The only closed invariant subspaces V ⊆ H are {0} and H.
2. Any vector φ ∈ H is cyclic.
The first version can als be fromulated as “there are no non-trivial projectors ∈ H that commute with
π(A)∀A ∈ A.
4.4
Sum and tensor product of representations
The concepts of direct sum and tensor product can be introduced more abstractly than what we do here.
We, in both cases, assume that our Hilbert spaces are equipped with a denumerable bases {|ii} (it is a
separable Hilbert space) and use these for the following definition.
Definition: Direct sum of Hilbert spaces
Let J and K be Hilbert spaces with the bases {|ji} and {|ki}. Then the space spanned by the basis
|ii⊕ = |ji i ⊕ |ki i
(48)
|mi⊕ + |ni⊕ = |jm i ⊕ |km i + |jn i ⊕ |kn i = |jm + jn i ⊕ |km + kn i
(49)
with the addition rules
and the scalar product
hm|ni⊕ = hjm |jn i + hkm |kn i
(50)
and analogously for the multiplication by a scalar α ∈ C is a Hilbert space. Show completeness
Definition: Tensor product of Hilbert spaces
Let J and K be Hilbert spaces with the bases {|ji} and {|ki}. Then the space spanned by the basis
|ii⊗ = |ji i ⊗ |ki i
27
(51)
and the scalar product
hm|ni⊕ = hjm |jn ihkm |kn i
(52)
and analogously for the multiplication by a scalar α ∈ C is a pre-Hilbert space. Its completion
span(|ji ⊗ |ki) = J ⊗ K = H
(53)
and is called the tensor product of J with K.
4.4.1
Illustration
Definition: Direct sum of operators
Let B ∈ B(J ) and C ∈ B(K) be operators on two Hilbert spaces. Their direct sum is defined by its
action on the basis functions:
B(J ⊕ K) 3 C|ii =: (B ⊕ C)(|ji i ⊕ |ji i) = (B|ji i) ⊕ (C|ki i).
(54)
As the operators are bounded (=continuous), this can be extended to a definition on the complete space.
Definition: Tensor product of operators
Let B ∈ B(J ) and C ∈ B(K) be operators on two Hilbert spaces. Their tensor product is defined by
its action on the basis functions:
B(J ⊗ K) 3 C|ii =: (B ⊗ C)(|ji i ⊗ |ji i) = (B|ji i) ⊗ (C|ki i).
(55)
As the operators are bounded (=continuous), this can be extended to a definition on the complete space.
Definition: Direct sum of representations
Let π1 and π2 be two representations of the same algebra on the Hilbert spaces H1 and H2 . The direct
sum of the representations is defined by
π1 ⊕ π2 = π : A → B(H1 ⊕ H2 ) : π(A) = π1 (A) ⊕ π2 (A)
(56)
Direct sums of representations are the prototypical form a redicible representations.
4.5
Illustration
• Draw vectors and matrices
• Point out the important differences!
• Remind of tensor product; as to direct sum: archetypical form of reducible representation
4.6
Problems
Problem 4.15: Irreducibility Let π be a representation of an algebra A on a Hilbert space H.
Show that the two definitions of irreducibility are equivalent:
1. The only closed invariant subspaces under π are H and {0}.
2. Any non-zero vector of H is cyclic.
28
Problem 4.16: Positivity and boundedness
(a) Use the Gel’fand isomorphism to argue that A∗ A ≤ ||A∗ A||1
(b) Convince yourself, that for operators in Hilbert space the definition of positivity translates into
hψ|Bψi ≥ 0 ∀ψ ∈ H.
(c) Show that positivity is conserved in representation, i.e. A positive ⇒ π(A) positive
(d) Use the above to show that any representation π : A → B(H) is a continuous map. (Remember the
relation between continuous and bounded for linear maps.)
5
The GNS construction
It is easy to see that, given a C ∗ algebra (which in particular is a linear space) and a state, one can
construct a linear space with a hermitian sesquilinear form
hA|Bi =: ω(A∗ B).
(57)
show hermiticity, using positivity. Unfortunately, positive definiteness is missing: hA|Ai = 0 6⇒ A =
0. We never excluded that a given ω has 0 expectation
value for some observable A∗ A. We resolve this
p
∗
by removing these vectors with zero “ω-length” ω(A A) = 0.
To make this precise, we need to introduce a few concepts. First, we note that elements of the algebra
with ω-length 0 transmit this property by multiplication:
The left-sided ideal Nω
Denote the set of elements with zero length by
Nω = {A ∈ A|ω(A∗ A) = 0}.
(58)
Then for any N ∈ Nω we have ω(N ∗ A∗ AN ) ≤ ||A∗ A||ω(N ∗ N ) = 0, i.e
N ∈ N ⇒ AN ∈ Nω .
(59)
(One says: Nω is a left-sided ideal of A).
Proof.
• A∗ A < ||A∗ A||1 - through Gel’fand isomorphism, function is positive,
C ∈ A exists.
p
||A||2 − A∗ A =:
• then 0 < (CB)∗ CB = B ∗ C 2 B = B ∗ (||A||2 − A∗ A)B or B ∗ A∗ AB < ||A∗ A||B ∗ B.
• by positivity and linearity of states: 0 < ω(||A∗ A||B ∗ B−B ∗ A∗ AB) = ||A∗ A||ω(B ∗ B)−ω(B ∗ A∗ AB)
• N ∈ N , A ∈ A: ω(N ∗ A∗ AN ) ≤ ||A∗ A||ω(N ∗ N ) = 0
We can now define an algebra, from which N is “divided out” (the quotient algebra): the idea is that
if any two elements differ only by an element of ω-length 0, they are considered as equivalent, both being
the valid representatives of an equivalence class.
Definition: Quotient space
We denote by bB the sets
bB := {A ∈ A|∃N ∈ Nω : A = B + N }
The set of the equivalence classes b is denoted by A/Nω (the quotient space).
29
(60)
Equivalence class The relation
A ∼ A0 : A = A0 + N ∈ N
(61)
defines an equivalence relation
A
∼ A
A
∼ B⇔B∼A
A
∼ B,
B∼C⇒A∼C
The sets bB form equivalence classes, meaning
A ∼ B ⇔ bB = a A
(62)
Any representative B from the class is equally good. Note in particular that the 0 ∈ A/Nω corresponds
to Nω ⊂ A (it is a true subset, why?).
With this we can construct a pre-Hilbert space
Lemma
Let ω be a state, then the quotient space A/Nω with the scalar product
hbB |bB 0 i = ω(B ∗ B 0 )
(63)
is a pre-Hilbert space.
Proof. The only non-trivial thing to show is bB = 0 ⇒ hbB |bB i = 0. Assume bB 6= 0, i.e. B = A + N
with A 6∈ Nω :
ω(B ∗ B) = ω(A∗ A) +
ω(N ∗ A)
| {z }
|ω(N ∗ A)|2 ≤ω(N ∗ N )ω(A∗ A)
+ω(A∗ N ) + ω(N ∗ N ) > 0 = ω(A∗ A).
| {z }
=0
We know |ω(A∗ N )| ≤ ω(A∗ A)ω(N ∗ N ) = 0 and we get A/Nω 3 b 6= 0 ⇒ hb|bi =
6 0.
From this we can obtain a Hilbert space by including all limits w.r.t. to the ω-length into our space,
by operating in the closure A/Nω .
We can immediately define an action of our algebra on the vectors of this space: let b ∈ A/Nω and
B ∈ A a representative of b, i.e. b = bB . Then
π(A)|bi = |ci,
cAB = {C ∈ A|C = AB + N, N ∈ Nω }.
(64)
Strictly speaking this is only defined for the quotient algebra A/Nω , but by continuity we can extend it
to being defined also on its completion A/Nω .
5.1
5.1.1
Reducible and irreducible GNS representations
1 ∈ A/N is cyclic
Trivially so, as any A/N 3 b = bB = π(B)1.
5.1.2
Irrep ⇔ pure state
In the exercises you have shown that the GNS representation corresponding to a pure state is irreducible.
We do not prove that also the converse holds: if a GNS representation is irreducible, the corresponding
state is pure.
30
5.1.3
Irreps for abelian C ∗ algebras
For an abelian algebra, any irreducible representation must be one-dimensional. This intuitive, but not
completely obvious.
As the GNS representation from a pure state is irrep it is one-dimensional, i.e. it is equivalent to a
*-homomorphism into C. That implies that a pure state on an abelian algebra is a character. We had
used this for Gel’fand theorem. Note that for GNS we do not make any reference to Gel’fand, ie. the
argument is not circular.
Problem 5.17: Pure states and characters Accepting the fact that any irrep of an Abelian
C ∗ algebra is one-dimensional, show that pure states on Abelian C ∗ algebras are characters.
Hint: Evaluate for GNS h1|(πω (A)|1i) and note that in 1-d πω (A)|bi ∝ |bi.
5.1.4
Any representation is the (possibly infinite) sum of irreps
Suppose we have a representation in a separable Hilbert space H. Remember that any vector |ii ∈ H
defines a pure state. According to the previous statement, the selected state defines an irrep GNS, and
the GNS space is isomorphic to an invariant subspace of Hi ⊂ H. We can now pick a state |ji 6∈ Hi and
repeat the procedure to obtain an new irrep on Hj ∩ Hi = {0} (being irreps, if they share one vector
they need to be identical). As the Hilbert space is separable, the iteration of the procedure exhausts the
complete space.
Each of the irreps is isomorphic to the corresponding irrep GNS.
Problem 5.18: GNS representation
(a) Find at least one cyclic vector for the GNS representation
Show that a pure state ω produces a irreducible GNS representation πω . We denote by G the GNS space,
by B(G) the set of all bounded operators on it, and by |Ωi ∈ G the vector that corresponds to 1 ∈ A.
Follow the steps:
(b) Show that ω pure ⇔ any positive linear functional µ with µ ≤ ω is µ = λω.
Hint: ⇒: convince yourself that ω = µ + (ω − µ) is a convex linear combination of two states
Hint: ⇐: construct µ 6= λω, 0 < µ < ω from ω = αω1 + (1 − α)ω2 . E.g., imagine a density matrix
ρ with two entries on the diagonal and construct a matrix that is not proportional to that one, but
still smaller.
(c) Let P 2 = P, P = P ∗ ∈ B(G) be a projector onto an invariant subspace. Convince yourself that
P π(A) = π(A)P ∀A.
(d) Show that µ(A) : A → hP Ω|π(A)P Ωi is a positive linear functional on A with the property
µ(C ∗ C) ≤ ω(C ∗ C), ∀C ∈ A, i.e. µ ≤ ω.
(e) Consider the matrix elements hπ(A)Ω|P π(B)Ωi and determine the possible values of λ of µ = λω.
Conclude that P = 1 or = 0.
Interestingly, also the converse is true, i.e. a GNS representation for a state ω is irreducible if and only
if ω is pure.
Problem 5.19: Convergence of operators Let B(H) denote the space of bounded operators
on a finite-dimensional Hilbert space H and let {Bn } denote a sequence of operators. Show that the
three following definitions of convergence are equivalent
1. Norm-convergence: Bn → B ⇔ ||Bn − B|| → 0 (Remember: ||B|| = sup||ψ||=1 ||Bψ||)
s
2. “Strong” convergence: Bn → B ⇔ ||Bn ψ − Bψ|| → 0∀ψ ∈ H
31
3. “Weak” convergence: Bn * B ⇔ hφ|(Bn − B)ψi → 0∀φ, ψ ∈ H
(Note the notation by *).
On infinite-dimensional Hilbert spaces this is not the case, there is only the implication norm-convergence
⇒ strong convergence ⇒ weak convergence.
s
(a) Using H = l2 , i.e. the space of infinite length vectors, construct Bn → B, but Bn 6→ B
s
(b) Construct Bn * B, but Bn →
6 B.
Hint: In both cases the trick is to let escape the maps Bn to ever new directions, e.g. by a sequence of
operators that connect ever new pairs of entries etc.
32
6
Towards the spectral theorem for bounded operators
We will next derive the spectral theorem for normal operators in Hilbert space. The steps for this
reasoning are as follows:
1. By Gelfand isomorphism, the algebra spanned by the normal operator A is isomorphic to the
algebra of bounded on functions σ(A). In particular, the operator A corresponds to the function
fA (a) = a.
2. We will show that any state ωi defines an integration measure µi (a) on σ(A) by identifying the
functionals ω with the integral
Z
ω(P (A)) =
dµi (a)fP (A) (a)
σ(A)
(By its boundedness, the ω extends to the complete algebra of bounded functions, not only the
polynomials, B.L.T. theorem to be proven).
3. The GNS Hilbert space Hi is then isomorphic to the Hilbert space L2 (dµi , σ(A)).
4. As the Hilbert space H of any representation is isomorphic to a direct sum of GNS-representations
Hi for different states ωi , it is also isomorphic to a direct sum of L2 (dµi , σ(A))
The last statement is what we know as the spectral theorem: there is a invertible, norm-conserving map
from any Hilbert space H where a (bounded) operator is defined to a direct sum of Hilbert spaces Hi ,
where the operator A appears as multiplication by the spectral value. A given subset τ ⊂ σ(A) of the
spectrum may appear with measure µi (τ ) > 0 in several of the representations Hi : this is what you first
encounter by the name of a “degenerate” eigenvalue.
7
Elements of measure theory
(A representation of this can be found in Teschl: Mathematical Methods in Quantum Mechanic).
Loosely speaking a measure on a set X is a function that assigns to any reasonable subset of X a
real number in a way that is additive, i.e. the measures of disjoint sets just add up. “Reasonable” needs
to be qualified in the more precise concept of a Σ-algebra. Once this is given, one can use the concept of
the Lebesgue integral to define integrals over the set.
7.1
Σ-algebra
Definition 1. A Σ-algebra is a set A of subsets of a given set X such that
1. X ∈ A
2. A is closed under countable many unions and intersections, and under complements.
7.2
Borel sets
For a given topology on a space X (i.e. a definition of the open sets, e.g. open balls on a metric space),
one can construct a Σ-algebra by just accepting all subsets of X that can result from countably many
set operations. This algebra is called the Borel Σ-algebra and its elements are the Borel sets.
It is useful to consider the following examples of which sets are Borel-sets for the real axis (with the
standard topology)
1. Closed sets
2. Single points as a special case of a closed set or also as
33
T∞
n=1 (x
− n1 , x + n1 )
3. The Cantor set is obtained by the following procedure: From the interval C0 = [0, 1], remove the
middle third, i.e. C1 = [0, 1/3] ∪ [2/3, 1]. Iterate by applying the analogous procedure to the the
two subintervals, take it to infinity. The resulting set has Lebesgue-measure zero and it has a
non-countable number of points.
Problem 7.20: C onstruct the Cantor set and show the above properties.
Problem 7.21: S how that the Cantor set is Borel and that a function on the cantor set is continuous
although its derivative = 0 everywhere
Problem 7.22: Cantor set The Cantor set is obtained by recursively removing the open invervals
(1/3, 2/3) from the compact interval [0, 1], then again the center third from [0, 1/3] and [2/3, 1] etc.
(a) Show that C is a Borel set with Lebesgue measure λ(C) = 0.
Hint: Study its complement
(b) Show that C = limn→∞ Cn , where Cn is constructed by recursion
Cn+1 =
2 1
1
Cn ∪ [ + Cn ]
3
3 3
(c) Show that the Cantor set is not countable
Hint: Write the points in the Cantor set as ternary (base 3) numbers, show that it contains all
numbers with digits 0 and 2).
(d) Show that the Cantor set does not contain any isolated points, i.e. for any point x in C and any
> 0 there is y ∈ C, such that |x − y| < .
7.3
Measure and measurable space
Definition 2. Let Σ denote a Σ-algebra of X. Then a (positive) measure µ is a map Σ → R with the
property
1. µ(E) ≥ 0
∀E ∈ Σ
2. µ(∅) = 0
3. For countably many Ei that are mutually disjoint Ei ∩ Ej = ∅, it holds
[
X
µ( Ei ) =
µ(Ei )
i
i
The pair (X, Σ) is called a measurable space, Σ are the measurable sets. If a concrete measure is
defined we call the triplet (X, Σ, µ) a measure space.
7.4
A few definitions for the Lebesgue integral (reminder)
Measurable function Let (X, Σ) be a measurable space. A function f : X → R is measurable if
the pre-image (inverse image) {x ∈ X|f (x) > t ∈ R} is in Σ. As we are for now mostly dealing with
continuous functions, this is trivially the case.
Simple function is a finite linear combination of characteristic functions χMk of measurable sets Mk
X
s=
ak χMk
(65)
k
A little caution is required if terms can be infinite. However, we will be dealing with finite measures
µ(X) < ∞, where this cannot happen.
34
Integral of a positive simple function
Z X
ak χMk =
k
X
ak µ(Mk )
Integral of a positive measurable function
Z
Z
f dµ := sup
sdµ, s simple
X
(66)
k
(67)
s<f
Integrals of non-positive and complex-valued functions are obtained by taking the function
apart into positive and negative parts f = f+ + f− and treating the two parts separately. Similarly,
integrals of complex valued functions are treated by integrating real and imaginary parts separately.
7.5
Lebesgue measure
A particular form measure on the Borel sets of the real axis is the Lebesgue measure used in standard
integration theory. There, we assign to an (open) interval its length as its measure. This carries over to
all intervals and in fact to all sets that can be written as unions of intervals by the additivity. On Borel
sets that procedure is guaranteed to work (no proof given here).
On this basis, we can proof
1. Any countable union of points has Lebesgue measure 0.
2. The Cantor set, although not countable, still has measure 0.
7.6
Absolutely continuous
A characteristic of physics is the existence of discrete and (absolutely) continuous spectrum of an observable. The typical physical example of absolutely continous spectrum is the spectrum of kinetic energy
or any unbound part of the spectrum. We were instructed that wavepackets of the free motion decay
by “wave packet spreading”, eventually to arbitrarily far distances. This is in fact a general property
of packes on the ac spectrum, which we will prove later. In contrast, wavepackets on the discrete spectrum remain confined, possibly undergoing “revivals”. This close association of basis characteristics with
phyiscal gives the distinction of point and ac spectrum its physical relevance.
Mathematically, there can be more than these two kinds, let us call the rest the “singular” spectrum.
In fact, such a spectrum may even appear in physical system such infinitely extended, non-periodic
systems, such as created by Penrose tiling of a plane. (See, e.g. a review by Yaron Last on Barry Simon’s
contributions).
The notions can be made precise definition and the subsequent theorem.
Definition 3. A measure µ is called absolutely continuous (a.c.) w.r.t. the Lebesgue measure λ on
the real axis, if λ(A) = 0 ⇒ µ(A) = 0.
7.6.1
Examples
1. An easy example for an absolutely continuous measure is what we usually do when we “change the
integration coordinate”:
dµ
dµ(x) =
dx,
dx
where dµ/dx is a non-negative function of x. More closely to our present notation we should write
Z
µ(E) =
f dλ
(68)
E
In fact, this is already the most general form of an absolutely continuous measure.
35
2. Clearly, point measures are not a.c.
3. Interestingly, also any measure that is non-zero on a Cantor set is not a.c.: it is neither a.c., nor
discrete (countably many points), it is the “other” mentioned above.
Two different measures µ, ν can be mutually singular. This means that one can find disjoint sets
M ∩ M = ∅, M ∪ N = X such that
µ(N ) = ν(M ) = 0,
(69)
which also implies µ(M ) = ν(N ) = 1. Loosely speaking: the two measures do not share any subsets
where both are non-zero.
Now we can formulate
Theorem 1. Lebesgue’s decomposition theorem. Any Borel measure ν on the real line can be decomposed
into
ν = νac + νp + νsing
(70)
with the three parts
νac
νp
νsing
absolutely continuous
discrete
singular,
which are mutually singular and νp and νsing are both singular w.r.t. the Lebesgue measure.
7.7
Signed measures and complex valued measures
One can abandon the positivity requirement on the measure and rather admit any real or even complex
number to be the measure of Borel sets. The most obvious example is to integrate with a signed or even
complex “weight” function. The property that should be maintained is the additivity. The measure can
be taken apart into imaginary and real parts, finally into the positive and negative parts. For each of
these at most 4 parts, integration can be defined by the Lebesgue procedure.
7.8
Projector valued measures
One can make another turn of the screw and assign to the measurable sets E (self-adjoint) projectors
P (E) onto some subspace of a Hilbert space H. It should have the properties
1. π(X) = 1, i.e. the measure of the complete measure space is the identity on the Hilbert space.
2. For any two functions φ, χ ∈ H, the map E → hφ|P (E)χi should be a complex valued measure.
Problem 7.23: I t follows from the above that the projection values measure of two disjoint sets
E∩F =∅
(71)
P (E)P (F ) = 0
(72)
are orthogonal
and more that the measures of any two sets commute
P (E)P (F ) = P (E ∩ F ) = P (F )P (E).
We will use such measures to formulate the spectral theorem more precisely.
Integrals using projection values measures have as their results operators.
36
(73)
7.9
Positive operator valued measures
The idea can be carried even further, although this will not be relevant for us: instead of projectors,
on can assign positive operators to the Borel sets such that they add up over all X to unity, but the
operators on disjoint sets do not need to be orthogonal.
7.10
Measures and continuous functionals
It turns out that positive linear functionals on continuous functions (at least on compact sets) all have
the form of an integral for the Borel Σ-algebra.
Theorem 2. (Riesz-Markov) Let X be a compact Hausdorff space. For any positive linear functional ω
on C(X) there is a unique (finite) Borel measure µ on X with
Z
l(f ) = dµf
(74)
Hausdorff space (separated space): “different points are separated”, any two distinct elements from
the space have disjoint neighborhoods.
Finite measure: the measure of the complete space is finite µ(X) < ∞.
Of course, the measure is not usually the Lebesgue measure.
We see where this is going: our states define a measure on the character set, which is — we have used
it before without proof — a compact set and a separated space.
Problem 7.24: Projection valued measures Let P be a projection valued measure.
(a) Show that for any two Borel sets E ∩ F = ∅ ⇒ P (E)P (F ) = 0.
Hint: Use additivity of the measure, the projector property P (E)2 = P (E) and study [P (E) +
P (F )]2 .
(b) From that, conclude that the measures of any two sets commutes P (E)P (F ) = P (F )P (E).
Hint: Split the sets into E = E 0 ∪ S, S = E ∩ F etc.
Problem 7.25: Operators as integrals with projection valued measures
(a) Show, formally (i.e. without considering the existence of a limit), that any hermitian bounded
operator on a Hilbert space H can be written in the form of an integral with a projection valued
measure on its spectrum
Z
A=
αdP (α)
(75)
σ(A)
Hint: Construct the projection valued measure from the characteristic functions for the half-open
intervals [α, ∞) at each L2 (dµi , σ(A)), use the isomorphism with the irrep subspace Hi to obtain a
projection valued measure Pi ([α, ∞)) on Hi .
(b) Show, formally, that any function of A can be written as
Z
f (A) =
f (α)dP (α)
(76)
σ(A)
8
The spectral theorem for normal operators
We can now go through the program for the above derivation of the spectral theorem. We consider the
commutative C ∗ -algebra spanned by {1, A, A∗ }.
37
8.1
Gel’fand isomorphism
This has all been discussed at length: the algebra is nothing but the continuous functions on the spectrum.
8.2
A state defines a measure on σ(A)
A state — originally a positive linear functional on the algebra — can also be understood as a positive
linear functional on the continuous functions C(X) over the character set X of the commutative algebra
by defining
ωX (fC ) := ω(C),
(77)
where C is an element from the Algebra and fC ∈ C(X) the corresponding function.
Similarly, there is a continuous, positive linear functional on the continuous functions on σ(A): ωσ :
C(σ(A)) → C, ωσ (fC ) = ω(C).
The spectrum is a a compact set: it is a closed (the resolvent set is open) and bounded |a| ≤ ||A||
subset of the complex numbers. Therefore we can apply Riesz-Markov to see that there is a measure on
σ(A) such that
Z
Z
ωσ (fσ ) =
dµσ fσ =
σ(A)
dµ(a)fσ (a).
(78)
σ(A)
(The last being only a convention of notation.)
8.3
Construction of the L2 (dµi , σ(A))
This is always the same trick: we do it for polynomials and then use the available norm to complete
the space to make it Hilbert. Clearly, all polynomials on σ(A) appear as vectors of the GNS. We define
an isomorphism between the vectors |P (A)i = P (π(A))|Ωi of the GNS representation and the functions
P (a) on σ(A). By the isomorphism, these are ∈ L2 (dµi , σ(A)). The isomorphism extends to the normcompletion of either space by a standard procedure.
2
In this way we have obtained an isomorphism between the
LL (dµi , σ(A) and our irreps on Hi and,
putting then all together, the original representation on H = i Hi .
8.4
Spectral theorem by projection valued measures
You will show in the exercises that one can formally write any hermitian operator as as an integral
Z
πω (A) =
αdPω (α)
(79)
σ(A)
with a projection valued measure on the spectrum Pω . We can restrict ourselves to irreps, where ω refers
to a given pure state that characterizes our irreducible representation. The difference to the previous is
that we directly construct πω (A) ∈ B(H), rather than using the isomorphism: the map is already built
into the construction. This technique will be used to construct the spectral representation for unbounded
operators, where we can no longer refer to the GNS constructions, that was all based on the C ∗ properties,
in particular the finite norm of the operator.
The integral is to be understood in the Lebesgue sense as
Z
X
αdPi (α) = sup
bi Pω (Mi ),
(80)
σ(A)
s
where the supremum is over the simple functions s(a) ≤ a.
There are two points that one must take into consideration: first of all, we have not proven that we
actually can construct a projector valued measure from our state ω (except for the trivial projectors 0
and 1). We have our measure by identifying the action of ω on elements of the algebra with an integral,
but for elements in the algebra, which correspond the continuous functions c ∈ C(σ(A)). Projectors Pα
38
correspond to characteristic functions χ[α,∞) , which are in general discontinuous and therefor are not in
the algebra. We can reach them by a limiting process, but not in the sense of the supremum-norm (after
all, our algebra is closed in this norm). Secondly, it is not obvious whether the integral is well-defined.
For both points we will use Lebesgue’s theorem about dominated convergence. The theorem establishes convergence beyond the (supremum) norm-convergence using pointwise convergence instead:
Theorem 3. Dominated Convergence (Lebesgue) Let {fn } be a sequence of real-valued measurable
functions on a measure space (S, Σ, µ). Suppose that the sequence converges pointwise to a function f
and is dominated by some integrable function g in the sense that
|fn (x)| ≤ g(x)
for n all points x ∈ S. Then f is integrable and
Z
lim
|fn − f | dµ = 0
n→∞
and also
S
Z
fn dµ =
lim
n→∞
Z
S
f dµ
S
Now, it is easy to construct a squence of continous functions cn ∈ C(σ(A) that converges to a
characterisitic function χM , M ⊂ Σ for any given x, “pointwise”. Also, the theorem guarantees that
the integral exists. Then χM exists as a µω -measurable function in L2 (dµω , σ(A)) and with that we can
define the projector-valued measure through
Z
hφ|P (Mi )ψi := φ∗ (a)χMi (a)ψ(a)dµω (a).
The same reasoning can be applied to show that any positive function, then any real, finally any
complex-valued function of a normal bounded operator can be defined as
Z
f (A) :=
f (α)dP (α)
(81)
σ(A)
We will come back to these ideas after the introduction of unbounded operators.
8.5
Quantum habits
In bra-ket notation of quantum mechanics, the above is
Z
P ([α, β])
β
=
Z
β
dµac (γ)|γihγ| +
α
Z β
=
dµp (g)|gihg|
(82)
α
dµac (γ)|γihγ| +
α
X
|gi ihgj |
(83)
gi ∈[α,β]∩σp (A)
(note that nobody usually considers any singular spectrum). The |γihγ| of σac are something like the
derivative of a projection operator, formally
dP (γ) =
dP
dγ
dγ
(84)
As the derivative of a step-function,
it has δ-character, and as you always suspected, the continuum
p
functions are something like “ δ(γ − γ 0 )”. Note however, that while we have met all instruments to
make things like dP (γ) household commodities, we have not remotely introduced the mathematics to
39
safely operate with the |γi. In fact, we really never need to do this, except for convenient formal
manipulations.
Problem 8.26: Limiting processes and measures Measures behave rather naturally under
set operations and in limiting processes.
(a) Let (X, Σ, µ) be a measured space and Y ⊂ X a measurable set. Show that
ν(A) := µ(A ∩ Y )
(85)
defines a measure on X.
(b) Show that A ⊆ B ⇒ µ(A) ≤ µ(B) (monotonicity)
Let us now define limits of sequences of sets as follows
An % A : An ⊆ An+1 and
[
An = A
(86)
An = A
(87)
n
and
An & A : An ⊇ An+1 and
\
n
(c) If An % A then µ(An ) → µ(A) (continuity from above)
Sn
Hint: Construct a series of distjoint sets such that m=1 Bm = An and compute the measures of
if the series.
(d) If An & A then µ(An ) → µ(A) (continuity from below)
Hint: Use the complement
40
9
9.1
Bounded and unbounded operators in Hilbert space
Linear operator
Let H be a separable Hilbert space. A linear operator is a map
A : D(A) → H
(88)
where D(A) is a linear subspace of H called the domain of A. A is called bounded if
9.2
||A|| := sup ||Aψ|| =
sup
||ψ||=1
||φ||=||ψ||=1
|hφ|Aψi| < ∞.
(89)
B.L.T. theorem (bounded line transformation)
Let A : D(A) ⊂ X → Y be a bounded linear operator and let Y be a Banach space. If D(A) is dense,
there a unique (continuous) extension of A to all of X, which has the same operator norm.
Problem 9.27: Proof the B.L.T. theorem for bounded operators in Hilbert space, which
says, that any bounded operator A with a dense domain D(A) can be uniquely extended to an continuous
operator A on all of D(A) = H with the same norm.
(a) Construct such an extension explicitly by finding a “natural” map for those elements, that are not
in the dense domain, and make sure you got a unique definition.
(b) Crosscheck linearity for the newly defined points of the operator.
(c) What is the reason the norms must agree?
(d) Explicitly convince yourself of the uniqueness.
9.2.1
Multiplication operator on L2 (dx, R)
Multiplication by a measurable function can be defined as an operator with a dense domain D(A): Let
(Af )(x) := A(x)f (x),
D(A) = {f ∈ L2 (dx, R)|Af ∈ L2 (dx, R)}
(90)
and A(x) is a measurable function.
For the proof the DA is dense, we use a procedure that is quite intuitive and will be used frequently
in the following: for any f ∈ H we construct a sequence of fn → f that is clearly in the domain. For that
we multiply f by a charactaristic functions that exclude regions where the integral would become too
large, the sets Ωn = {x||A(x)| ≤ n}. Ωn are Borel sets as we required A(X) to be measurable. Clearly
Ωn → R such that Ωm ⊂ Ωn for m ≤ n: one writes Ωn % R. The functions fn ∈ D(A) : Afn ∈ L2 (dx, R).
Also, ||fn − f || → 0 by Lebesgue dominated convergence, i.e. the fn are dense.
Next, one can easily see that A (considered as an operator) is bounded if the function A(x) is bounded.
(More precisely, it should be essentially bounded: it may be as large as it wants, but only on sets of measure
0. The correct way of capturing this idea there as a real number a the set Ia = {x||A(x)| > a} has measure
µ(Ia ) = 0.)
Problem 9.28: Bounded multiplication operator Let fA (x) be a measurable function
and A the multiplication operator on L2 (dx, R)
(Af )(x) = fA (x)ψ(x),
D(A) = {ψ ∈ L2 (dx, R)|Aψ ∈ L2 (dx, R)}
(91)
Show that A is bounded iff fA is essentially bounded, i.e. there is a real number a such that Ia =
{x| |fA (x)| > a} has measure 0.
41
(a) Show that ||A|| ≤ ||fA ||∞ , remember
||f ||∞ = inf{a|µ(Ia ) = 0}
(92)
(which adjusts the supremum-norm for the case of only essentially bounded functions).
(b) Show ||A|| <
6 ||fA ||∞ .
Hint: By contradiction: if we had ||A|| = c, but c < ||fA ||∞ we could construct a test function
φ ∈ H such that c < ||Aφ||that contradicts this.
It is good to remember why for an unbounded function the domain cannot be all of {L2 (dx, R)}:
for example, in the case case A(x) = x functions behaving as |f (x)| ∼ 1/|x|1/2+ , > 0 are certainly
∈ L2 , but the functions A(x)f (x) ∼ x/|x||1/2+ are not. Another example is A(x) = |x|−1 : any function
f ∈ L2 (dx, R) behaving as |x|1/2− near x = 0 will certainly have a diverging integral. By a similar
principle one can build functions f that are L2 for any A(x)f (x) that is not essentially bounded, i.e. that
exceeds any arbitrary bound on a set with non-zero measure.
Problem 9.29: Singular functions for unbounded multiplication operators Given
a positive measurable function A : R → R+ that is not essentially bounded. Construct at least one
function s ∈ L2 (dx, R) for which ||As|| = ∞.
9.3
Comments on the domain
A non-trivial dense domain is typical for unbounded operators. The existence of the domain also restricts
the freedom for algebraic operations involving unbounded operators. For example, the domain of the sum
of two operators D(A + B) is in general smaller then each of the domains of the pieces, we can choose
D(A + B) := D(A) ∩ D(B)
9.4
(93)
Quadratic form and polarization identity
It is interesting to see that we do not need to know all matrix elements of an operator. It suffices to know
all its “expectation values” hψ|Aψi. The map
qA : D(A) → C : qA (ψ) = hψ|Aψi
is called the quadratic form of the operator. We can construct all matrix elements of the operator from
its quadratic form by the polarization identity
hφ|Aψi =
9.4.1
1
[qA (φ + ψ) − qA (φ − ψ) + iqA (φ − iψ) − iqA (φ + iψ)]
4
Hermitian operator
A densely defined operator is called hermitian (= symmetric) if
hφ|Aψi = hAφ|ψi
∀φ, ψ ∈ D(A).
(94)
Lemma 1. A densely defined operator A is symmetric if and only if its quadratic form is real-valued.
Problem 9.30: Hermitian operators and quadratic forms Show that a densely defined
operator A is hermitian if and only if its quadratic form qA is real-valued.
Hint: Evaluate the quadratic form for a vector ψ + iφ
42
9.5
Adjoint operator
Let A be an operator on a Hilbert space H defined on a dense subset D(A) ⊂ H. Then the adjoint
operator A∗ is defined through
hφ|A∗ χi = hAφ|χi
∀φ ∈ D(A).
(95)
The domain D(A∗ ) are the vectors χ ∈ H for which sup||φ||=1 |hAφ|χi| < ∞. For bounded A, trivially
D(A∗ ) = H.
9.5.1
Comment on the domain D(A∗ )
The operator A∗ is well-defined and its domain D(A∗ ) is determined once we have chosen D(A). Note
that in general D(A∗ ) 6= D(A∗ ). It may not even be dense, let alone all of H, or anything useful at all.
Example
We can construct a δ-type operator on L2 (dx, [−1, 1]) as
Aφ = φ(0), D(A) = φ continuous
(96)
The χ ∈ D(A) means
Z
1
sup
χ∈D(A)
χ∗ (x)
−1
φ(0)
< ∞.
||φ||
(97)
As we can construct continuous L2 functions that are arbitrarily large at x = 0 (e.g. a δ-like sequence
of hat-functions), χ must be orthogonal to the constant function, i.e. the 1-dimensional subspace of
constant functions gλ (x) ≡ λ 6∈ D(A∗ ).
9.6
Range and kernel
One can see that for densely defined A,
ran(A)⊥ = ker(A∗ ) :
(98)
0 = hA∗ χ|φi = hχ|Aφi
(99)
Let χ ∈ ker(A∗ ), i.e. A∗ χ = 0
which means χ ⊥ Aφ. On the other hand χ ⊥ Aφ implies the scalar product = 0 for a dense set of φ, i.e.
A∗ χ = 0.
9.7
Self-adjoint operators
The main technical distinction in the transition from bounded hermitian to self-adjoint operators is that
not only the action of A and A∗ on some φ ∈ D(A) ∩ D(A∗ ) must be identical, but that it must be
possible to define D(A) such that also D(A) = D(A∗ ), only then we can write A∗ = A.
Any bounded hermitian operator can be made self-adjoint by extending its domain to D(A) = H,
which is always possible by B.L.T.
For a hermitian operator
hφ|Aφi = hAφ|φi < ∞
∀φ ∈ D(A).
Therefore, for a hermitian operator A, D(A∗ ) ⊇ D(A), we use the notation A∗ ⊇ A.
Note that A∗ is not in general symmetric(=hermitian), even if A is symmetric.
43
(100)
9.7.1
Multiplication operator
It is interesting to discuss this on the simple example of our multiplication operator. Let a multiplication
operator A be as above and let us define
à : (Ãh)(x) = A∗ (x)h(x), D(Ã) = {f ||Af || < ∞}
(101)
Note that here we have (plausibly) chosen the domain D(Ã) and clearly D(Ã) = D(A). But is à = A∗ ?
For the adjoint we cannot choose the domain, it is defined once A is fully defined. It is not a priori
obvious that there cannot be a h 6∈ D(Ã) such that |hh|Afn i| < ∞ ∀fn ∈ D(A). To show that this cannot
happen, let us assume h ∈ D(A∗ ). Then ∃g = A∗ h such that
hh|Afn i − hg|fn i = 0 ∀fn ∈ D(A).
(102)
Now we use again the trick with restricting the integrations to a subset Ωn of R with finite measure
Ωn = {x| |A(x)| < n}, in which case fn = χΩn f ∈ D(A) for f ∈ L2 , therefore also
Z
h∗ (x)A(x)fn (x) − (A∗ h)(x)fn (x)dx =
(103)
Z
[h(x)A∗ (x) − (A∗ h)(x)]∗ χΩn f (x)dx = 0 ∀f ∈ L2 ,
(104)
As Ωn ↑ R, this shows that (except possibly for sets of measure 0)
A∗ (x)h(x) = (A∗ h)(x).
on any h(x) ∈ D(A∗ ) the adjoint acts as multiplication by the complex conjugate function. By construction of the adjoint, ||A∗ h|| < ∞ and therefore h ∈ D(Ã).
It follows immediately, that real-valued multiplication
operators for measurable functions A(x) are
self-adjoint, if we choose the domain as D(A) = {f ||Af || < ∞}.
9.8
Closed operator
If we have an densely defined unbounded operator we loose continuity. One typical way how unbounded
operators can behave is seen with the multiplication operators: as we approach with our functions fn ∈ L2
the places where A(x) diverges, the norms diverge ||Afn ||. Somehow one feels that this is a rather benign,
at least well understood failure of convergence. Somehow we are warned about what we are running into
as we approach the fringes of our domain. I we watch our steps and walk more slowly as ||Afn || grows,
nothing bad will happen. Contrast this with the situation where our domain just has a hole and nothing
near the hole indicates that there is a problem, like a δ-function.
The former case of unbounded operators is a very important one and justifies the following definitions
Definition 4. The graph ΓA of an operator A is the set of pairs ΓA = {(ψ, Aψ)|ψ ∈ D(A)}.
Definition 5. The graph-norm of ψ ∈ D(A) for A is ||ψ||A := ||ψ|| + ||Aψ||.
Definition 6. An operator A is called closed, if D(A) is closed w.r.t. the graph norm || · ||A .
Note that, if the graph ΓA of A is closed as p
a subset of H ⊕ H, the operator is closed: although the
graph norm ||φ||A is not identical to the norm ||φ||2 + ||Aχ||2 defined on H ⊕ H, the two norms are
equvialent: if a sequence (φn , Aφn ) ∈ ΓA ⊂ H ⊕ H Cauchy in either norm, it is also Cauchy in the other:
(||ψn || + ||Aψn ||)2 → 0 ⇔ ||ψn ||2 + ||Aψm ||2 → 0
any convergent sequence of Ψn = (ψn , Aψn ) ∈ ΓA converges, if ||ψn ||A = ||ψn || + ||Aψn || converges. If we
know that ΓA is closed w.r.t. the either norm, it is closed w.r.t. to the graph norm, i.e. the operator is
closed.
Closed operators are such that, if both a sequence ψn and the sequence Aψn converges, we know that
ψn → ψ ∈ D(A). Typical cases where this can fail are δ-function like objects.
44
Lemma 2. The adjoint operator A∗ of any A is closed.
This can be shown by a neat trick as follows.
Proof. In H ⊕ H we consider the unitary map J : (φ, χ) → (−χ, φ). (for real vectors, this map is like an
orthogonal rotation in that
h(φ, χ)|J(φ, χ)i = hφ, χ| − χ, φi = −hφ|χi + hχ|φi.)
The graph of the adjoint is the set of pairs of χ, η ∈ H ⊕ H such that hχ, η|J(φ, Aφ)i = 0, i.e. hχ|Aφi =
hη|φi for all φ ∈ D(A). That means graph the (χ, A∗ χ), χ ∈ D(A∗ ) consists of the pairs that are
orthogonal to the subspaces J(φ, Aφ), φ ∈ D(A). We use that the orthogonal subspace of any subspace
is closed. Therefore the graph is closed w.r.t. to the H ⊕ H norm, which on the graph is equivalent to
the graph norm.
The discussion above suggests that operators that are defined everywhere and are closed must be
bounded. That is indeed the case:
Problem 9.31: Closed graph theorem Let A : H → H be a linear map that is defined on all
H. A is bounded iff Γ(A) is closed.
Problem 9.32: Closed operator - counter-example Show that the operator (Aφ)(x) =
φ(0) on D(A) = {φ ∈ L2 (dx, [−1, 1]), φ continuous} is not closed.
Hint: Construct a series such that (φn , Aφn ) = (φn , 1), where φn → φ = 0 and compare the limit of the
pairs with the pair at φ, Aφ.
9.9
Normal operator
We would also like think that of our multiplication operator as normal in the sense AA∗ = A∗ A. But
here we are in trouble, as even for real valued functions we do not a priory know what would be the
operator A∗ A: what is the domain? This is not a trivial question, as, e.g., the domain for A(x) = x
definitely differs from the domain for A2 (x) = x2 . We need to amend our previous definition “normal”:
Definition 7. An operator is called normal if D(A) = D(A∗ ) and ||Af || = ||A∗ f || for f ∈ D(A).
Problem 9.33: Normal unbounded operators Bounded normal operators are defined by
A∗ A = AA∗ . Because of the problems with defining the domain, for unbounded operators one calls
A normal, if D(A) = D(A∗ ) and ||Af || = ||A∗ f || ∀f ∈ D(A). Show that for bounded operators this
coincides with our previous definition.
Problem 9.34: Normal operators are closed Show that.
Hint: Use the lemma that A∗ is closed (will be proven in the lecture). The graph norms can be related
using the definition of normality.
9.9.1
The domain of self-adjoint operators is maximal
If we have defined an operator on a dense set and if we know it is self-adjoint, the domain is fixed and
we cannot extend it or reduce it without destroying self-adjointness.
Suppose there is B ∗ = B ⊇ A = A∗ . For any g ∈ D(B ∗ ), by definition |hg|Ahi| < ∞∀h ∈ D(B) ⊇
D(A), so we know g ∈ D(A∗ ), i.e. D(A∗ ) ⊇ D(B ∗ ). As we assume that D(A) = D(A∗ ) as well as
D(B ∗ ) = D(B), we have
D(A∗ ) ⊇ D(B ∗ ) = D(B) ⊇ D(A) = D(A∗ )
45
Caution this does not mean that all operators that are symmetric (but not self-adjoint), i.e. A∗ ⊃ A
can only be extended in one single way. This is definitely not the case: what you get to know as the
“δ-potential” is obtained by a particular extension of the kinetic energy operator, functions ψ(0) 6= 0
are removed from the domain, which destroys self-adjointness. From this point, different extensions are
possible, one of them being the “∆ + δ(x)”.
9.9.2
Closure of a symmetric operator
It is often easy to show that an operator is symmetric. It is often hard to show that it is self-adjoint.
Maximality of selfadjoint operators can help to proof an operator is self adjoint. The domain of the
adjoint operator is by definition maximal. For symmetric A it is larger than A∗ by definition of what
is symmetric: A∗ ⊇ A. The domain of the adjoint of the adjoint, D(A∗∗ ) is maximal. It contains all
vectors f such that A∗∗ f is a bounded linear functional. In particular, D(A∗∗ ) ⊇ D(A). When trying to
extend an operator A to a maximal symmetric (self adjoint) operator, one can use that D(A∗ ) contains
all vectors h for which hA∗ h|f i = hh|A∗∗ f i = hh|Af i exists, where f is from a dense set. We have the
relations
A∗ ⊇ A,
A∗ ⊆ A and also A∗∗ ⊇ A.
(105)
that is, A∗∗ is “smaller” (or equal) than A∗ , but it is “larger” (or equal) than A. Of course, A∗∗ f = Af
for f ∈ D(A). Therefore A∗∗ is an extension of A. It is called the closure A = A∗∗ . It can happen that
A = A∗ , in which case we call A essentially self-adjoint, implying that the self-adjoint extension of A
(by closure) is uniquely determined.
9.9.3
The importance of self-adjointness for quantum mechanics
Popularity of self-adjoint operators is due to the fact that they (and only they!) have strictly real
spectrum. This is convenient, if you want to directly associated a real number shown by some instrument
with the spectral value.
However, the true importance is that there is a one-to-one relation between self-adjointness and
unitary groups. Unitary groups strictly represent all differentiable symmetries and also, of course, the
time-evolution. No self-adjoint operator - no unitary evolution and loss of probability, conversely, if
there as a unitary time-evolution, there is a Hamiltonian that is self adjoint. Similar things hold for
any symmetry you might come up with: momentum, angular momentum, Lenz-Runge: the associated
conserved quantities MUST be represented by self-adjoint Hamiltonians. Naive manipulations with δ-like
objects can lead to leaks of probability into Nirvana.
9.9.4
Example: the spatial derivative
Definition of domains (for essential self-adjointness) is related to boundary conditions for differential
operators. Here we are at a point of great practical relevance: spectral properties, and with it the physical
meaning of an operator vastly differ with the domain that we assign to it. The following examples may
appear contrived or obvious in hindsight, but actually, they are very strong illustration of the kind of
problems associated with domain questions.
Let us compare the two operators on L2 (dx, [0, 2π])
(A0 f )(x)
D(A0 ) = {f ∈ C 1 ([0, 2π])|f (0) = f (2π) = 0}
= −i∂x f (x),
(106)
(Ap f )(x)
D(Ap ) = {f ∈ C 1 ([0, 2π])|f (0) = f (2π)}
Both operators are symmetric, domains are dense with D(A0 ) ⊂ D(Ap ). But only Ap is essentially
self-adjoint, while A0 is not.
Appealing to physics knowledge, we can anticipate why: derivatives are (infinitesimal) translations.
By exponentiating them, we should obtain macroscopic translations, but this cannot be for D(A0 ): we
cannot push a non-zero function across the boundary without loosing the norm.
46
Dirichlet boundary conditions, A0 We determine the domain of D(A∗ ). For any g(x) ∈ D(A∗0 )
there exists γ(x) ∈ L2 such that
Z 2π
Z 2π
dxg(x)(−i∂x f )(x) =
dxγ(x)f (x),
(107)
0
0
from which we conclude by partial integration that
Z 2π
Z x
0
dxf (x)[g(x) − i
dtγ(t)] = 0 ∀f ∈ D(A0 )
0
(108)
0
One can show (see problems) that this implies that g is “absolutely continuous”, i.e.
Z x
dtγ(t).
g(x) = g(0) +
(109)
0
We have established that A∗ g = γ = ∂x g such that γ ∈ L2 .
These are two necessary conditions, but together also sufficient and thus they define the domain. The
two properties define the “first Sobolev space” H 1 , which is the domain D(A∗ ).
Problem 9.35: Absolutely continuous Show that
Z
2π
0
Z
dxf (x)[g(x) − i
0
x
dtγ(t)] = 0 ∀f : f ∈ C 1 [0, 2π] and f (0) = f (2π) = 0}
0
implies that g(x) is “absolutely continuous”:
Z
g(x) = g(0) +
x
dtγ(t).
(110)
0
Note: In case you are wondering: there are functions c(x) that are continous, that are differentiable
almost everywhere (i.e. except on a set of measure zero), ∃∂x c, but are not absolutely continuous. A
famous example can be constructed using the Cantor set.
Problem 9.36: Absolutely continuous - counterexample We have seen that functions
f ∈ D(A∗0 ) of the adjoint of our differential operator
(A0 f )(x) = f 0 (x), D(A0 ) = {f ∈ C 1 [0, 2π]|f (0) = f (2π) = 0}
must be absolutely continuous, where C 1 denotes the functions that are continuously differentiable almost
everywhere.
(a) Construct a continuous function that is not absolutely continuous c(x).
Hint: Find a function with this property that is constant on the complement of the cantor set.
Extend it to a continuous function that is defined everywhere on [0, 1]. (Cantor function)
(b) Show that this function is not from the domain of D(A∗0 ).
Is A0 essentially self-adjoint? Here we can illustrate the concept of closure and essential self?
adjointness: for essentially self-adjoint operators the uniquely defined A∗0 = A0 gives the self-adjoint
extension of the original A0 . For the present case, we compute D(A). We choose any function f from
D(A0 ) ⊆ D(A∗ ),
0 = hg|A0 f i − hA∗0 g|f i = i(g ∗ (0)f (0) − g ∗ (2π)f (2π).
(111)
where the second identity is established by partial integration. But for g ∈ D(A∗0 ) we have no restriction
on the boundary values and in particular we can have g(0) 6= g(2π). Therefore, f (0) = f (2π) = 0: also
47
for the domain of the closure we must maintain the zero boundary conditions, i.e. A∗0 ⊃ A0 , A0 is not
essentially self-adjoint!
We do not have all info to extend A0 uniquely to a self-adjoint operator: we could, for example,
use periodic boundary conditions, which leads to Ap , but we could just as well keep the domain “nailed
down” at the boundaries, which would correspond to an extension by δ-functions. The resulting operator
is is no longer what we consider as a derivative ∂x .
Periodic boundary conditions, Ap Obviously Ap ⊃ A0 , therefore A∗p ⊆ A∗0 . We easily identify the
additional constraint on D(A∗p ) by integration by parts shows
0 = hg|Ap f i − hA∗p g|f i = [g ∗ (0) − g ∗ (2π)]f (0) ⇒ g(0) = g(2π).
(112)
D(A) = D(A∗∗ ) = {f | ac , ||f 0 || < ∞, f (0) = f (2π)},
(113)
which leads to
i.e. first Sobolev space with periodic boundary conditions. We have the same necessary and sufficient
conditions on D(A∗p ) as on D(Ap ), i.e. Ap is self-adjoint.
9.9.5
A criterion for self-adjointness
Lemma 3. Let A be hermitian. If there exists z ∈ C such that ran(A + z) = ran(A + z ∗ ) = H, then A
is self-adjoint.
Proof. We will show that any ψ ∈ D(A∗ ) is also in D(a). Let ψ ∈ D(A∗ ) and denote A∗ ψ = ψ̃. Since
ran(A + z ∗ ) = H, there is ϕ ∈ D(A) ⊆ D(A∗ ) such that (A + z ∗ )ϕ = ψ̃ + z ∗ ψ. Now
χ ∈ D(A),
hψ|(A + z)χi = hψ̃ + z ∗ ψ|χi = h(A + z ∗ )ϕ|χi = hϕ|(A + z)χi
As this holds for all χ ∈ D(A) and ran(A + z) = H, we conclude ψ = ϕ and thus ∈ D(A), i.e. any
ψ ∈ D(A∗ ) ⇒ ψ ∈ D(A) ⊆ D(A∗ ).
Conversely, we see that if A is hermitian, but not self-adjoint, any point of one of the complex halfplanes must be in the spectrum of A, i.e. A + z or A + z ∗ or both are not invertible.
9.9.6
Spectrum of a self-adjoint operator
As for elements of a C ∗ -algebra, we define the spectrum as the complement of the resolvent set. We ask,
more specifically, that the inverse exists as a bounded operator
ρ(A) = {z ∈ C|∃(A − z)−1 , ||(A − z)−1 || < ∞}
(114)
Using the above definition, we can see that the spectrum of a self-adjoint operator is strictly real:
1.
||(A − z)φ|| = ||(A − x)φ + iyφ|| ≥ |y| ||φ||
∀φ ∈ D(A).
(115)
i.e. ker(A − z) = {0}, it is invertible for all y 6= 0.
2. For self-adjoint operators, we also have ker(A∗ − z ∗ ) = {0}. As ran(A − z)⊥ = ker(A∗ − z ∗ ) we
know ran(A − z) is dense in H.
3. To see that it is not only dense, but all H, write ψ ∈ ran(A−z) as ψ = (A−z)φ, then ||ψ|| ≥ |y| ||φ||,
then
||(A − z)−1 ψ||
||φ||
≤
= |y|
(116)
||ψ||
|y| ||φ||
i.e. it is uniformly bounded for a dense subset, therefore also its supremum is bounded.
48
4. Finally, we know that for any ψ ∈ H there is a squence φn : Aφn → ψ. As self-adjoint operators
are closed, the limit φ, Aφ is in the graph, i.e. φ ∈ D(A).
Note: Self-adjointness is needed here, as for χ ∈ D(A∗ )\D(A) we cannot use the above argument for
bounding ||(A∗ − z ∗ )χ||, as we need to use hA∗ χ|χi = hχ|A∗ χi = hχ|Aχi ∀χ ∈ D(A∗ ), which we are not
allowed to do, if we only know our operator is symmetric. Note that the non-zeroness is not on a dense
subset is not sufficient to conclude ker = {0}.
Counter-example The operator D = −i∂x on [0, ∞) with zero boundary condition is hermitian.
However, its adjoint has the eigenvector e−x with eigenvalue −i. Therefore ker(D∗ + i) 6= 0 and ran(D −
i) 6= H (not even dense). Therefore D + i is not invertible and i ∈ σ(D).
• Is there an eigenfunction to the eigenvalue i? Does not need to be, for example also for x there is
no sensible “eigenfunction”.
• Is there a sequence of functions fn ∈ D(D) such that ||(D − i)fn || → 0? No, not in this case,
as actually the complete lower half plane is in the spectrum and we cannot approach the singular
point from non-singular points. We can only approach the boundaries of the resolvent set in this
way. For self-adjoint operators, any spectral point is a boundary point of the resolvent set.
• Again, in hind sight from Stone’s theorem, this is not surprising: the derivative would shift functions
to the left, but we cannot shift them to negative x, they are just not in our Hilbert space.
Problem 9.37: Hermitian, not self-adjoint
Note: There were a few misprints in this problem.
Let A be the operator on H = L2 (dx, [0, ∞))
(Af )(x) = (i∂x f )(x),
D(A) = {φ ∈ H|φ ∈ C 1 , φ(0) = 0}
(117)
(a) Show that A is hermitian
(b) Determine D(A∗ )
(c) Show that all z, =m (z) < 0 are eigenvalues of A∗ and determine the eigenfunctions.
(d) Conclude that σ(A) includes the whole upper half plane.
Note: We had only loosely introduced the resolvent set as the z where (A − z)−1 “exists as a
bounded operator”. It was implied that it is at least densly defined.
9.10
Quadratic forms
There is a feeling that, if only we know all possible expectation values of an operator, i.e. if we know
the associated quadratic form q, we know the operator. We had seen this in the “duality” between
“observables” and “states”. So, given a complete set of “expectation values” A(ψ), i.e. given a quadratic
form with a dense domain for ψ, can we find an operator A, such that E(ψ) = hψ|Aψi?
Yes, under some circumstances (Teschel Theorem 2.14). A general quadratic form derives from a
sesquilinear form on a Hilbert space s(φ, ψ) as q(ψ) := s(ψ, ψ). (It is the “diagonal” of the sesquilinear
form.) The connection to a self-adjoint operator can be established, if the quadratic form has a lower
bound, i.e. ∃γ ∈ R such that E(ψ) ≥ γ, ∀ψ ∈ Q(E) and if it is closed: Q is closed with the form norm
||ψ||q = ||(A − γ)ψ|| + ||ψ||.
Theorem 4. To every closed semi-bounded quadratic form q defined on Q there corresponds a unique
self-adjoint operator A such that for its quadratic form qA = q with Q = QA , where
ψ ∈ QA : qA (ψ) = hψ|Aψi < ∞
49
Proof. (find it in Teschl, Theorem 2.14)
Note: Typical Schrödinger operators are semi-bounded: there is a lowest energy. However, note that
~
in elementary quantum mechanics you meet the Hamiltonian for the Stark effect, with a potential ~r · E,
wich has no lower bound, with a lot of mathematical trouble as a consequence.
Note: A typical quadratic form that is not closed is the δ-potential
Z
δ(ψ) = dxψ(x)∗ δ(x)ψ(x) = |ψ(0)|2 :
(118)
tent-functions with their peak at x = 0 approach the 0-function, also the value of the quadratic form is
constant, i.e. it is a Cauchy-sequence under the form norm, but the limit under the form norm 6= 0.
10
The spectral theorem for self-adjoint operators
We have formulated the spectral theorem so far only for bounded normal operators. We will now extend
this to unbounded self-adjoint operators. We go through the following steps:
• We recall the spectral theorem for bounded operators in the formulation using projection valued
measures.
• Given any projection valued measure, show we can construct, for any decent (=measurable) function
f an operator P (f ). That operator will be normal. If f is unbounded, P (f ) is unbounded. The
main point is to get the domain right.
• We will circumvent the problems with unboundedness by considering the family of bounded operators
RA (z) = (A − z)−1 (Resolvent!).
• We will appeal to measure theory to state (without proof) that there exists a unique measure µψ (λ)
such that
Z
hψ|RA (z)ψi = (λ − z)−1 dµψ (λ)
(119)
R
• From this measure, one deduces a projection valued measure that allows formulating the spectral
theorem exactly as for bounded operators.
10.1
Bounded spectral theorem with projection valued measure on R
From now on we only treat self-adjoint operators, not the wider class of normal operators. We have
established that a commutative algebra of bounded hermitian operators that are part of a C ∗ algebra are
unitarily equivalent to the continuous functions on the spectrum. The spectral representation of any
such operator can be expressed as
Z
A=
λdP (λ)
(120)
σ(A)
for some projection valued measure P (λ) of the half-intervals (−∞, λ]. One can define the expectation
value
Z
hψ|P (λ)ψi =
χ(−∞,λ] (α)dµψ (λ).
(121)
R
In the original representation we had not Hilbert space given a priory, and the main difficulty was that
χ(−∞,λ] is not a continuous function, therefore it is not obvious, whether there exists such a P (λ).
With the Hilbert space given, one can prove the existence of P (λ) as a projector as follows.
50
Lemma 4. Let H be a Hilbert space and µψ be the measure on R corresponding to the state ωψ (A) =
hψ|Aψi on the C ∗ -algebra of the bounded operators on H. Then
Z
hψ|P (λ)ψi =
χ(−∞,λ] (α)dµψ (λ).
(122)
R
defines a projection operator on H.
Proof. We construct a sequence of continuous functions cn (t) that converge pointwise to the characteristic
function cn (t) ↑ χ(−∞,λ] (t). This lets us define the multiplication operators Cn and a sequence of quadratic
forms
Z
qn (ψ) = hψ|Cn ψi =
cn (t)dµψ (t).
(123)
R
For fixed ψ, qn (ψ) is Cauchy and we denote the limit by q(ψ). q is bounded from below as all q(ψ) =
lim qn (ψ) ≥ 0. It is closed: it is defined everywhere and the form norm ||ψ||q = ||ψ|| + q(ψ) < 2||ψ||
is equivalent to the norm on H. Therefore there exists an operator P (λ). By construction, P (λ) is the
strong limit of the Cn . It is bounded, as Cn ≤ 1. What remains to show is the projector property. As
we now know that all operators are bounded, showing the property in the strong limit is sufficient
hψ|(P 2 − P )ψi = hψ|(Cn2 − Cn )ψi + → 0
(124)
which can be shown by cutting from the integration the ranges Ωn = {t|cn (t)2 − cn (t) > }, for which
clearly µψ (Ωn ) → 0.
10.2
Resolution of identity P (λ)
We abstract the above properties of the P (λ) into the following definition:
Definition 8. A resolution of identity P (λ) := P ((−∞, λ]) with the properties
1. P (λ) is an (orthogonal) projection operator P 2 (λ) = P (λ) = P ∗ (λ).
2. P (λ1 ) ≤ P (λ2 ) for λ1 ≤ λ2
s
3. P (λn ) → P (λ) for λn ↓ λ (strong right-continuity)
s
s
4. P (λ) → 0 for λ → −∞ and P (λ) → 1 for λ → ∞ and
Lemma 5. To any projection valued measure there corresponds a resolution of identity. Conversely, any
resolution of identity uniquely defines a projection valued measure.
• Obviously, the projection valued measures of the intervals {(−∞, λ]} have these properties. Also,
this can be turned around: given a resolution of identity, there is a projection valued measure and
relation is unique (we do not proof this here).
• We need to restrict to right continuity as our measure may contain point-measures: thinking of a
characteristic function χ(−∞,λ] with σ({λ}) 6= 0: if we approach this from above, it is counted into
the measure all the time. If, in contrast, we approach λ from below, we would have a sudden jump
as we include the last point λ and cannot expect left-continuity.
10.3
Construction of operators from projection valued measures
We have discussed that the integral
Z
P (f ) :=
f (λ)dP (λ)
(125)
R
defines an operator for bounded continuous functions and for the characteristic function. Obviously, this
extends to bounded measurable functions (i.e. any function for which Lebesgue-integration works).
51
P (f ) for unbounded measurable f
We choose the domain
Z
Df = {ψ ∈ H| |f (λ)|2 dµψ (λ) < ∞},
(126)
R
where µψ (λ) = hψ|P (λ)ψi is a positive measure on R. It is easy to see that this is a linear subspace.
Problem 10.38: Show that Df is a (open) linear subspace and that it is dense.
Now, by ever the same trick, we construct a sequence of bounded functions converging to f by
multiplying by the characteristic functions
fn = χΩn f,
Ωn = {λ| ||f (λ)| < n}
(127)
which is Cauchy in L2 (dµψ , R) as f ∈ L2 . Then also P (λ)fn = ψn is Cauchy w.r.t. H: for any projector
P : ||P ψ|| < ||ψ||. Therefore we can define
P (f )ψ := lim P (fn )ψ,
n→∞
ψ ∈ Df .
(128)
Theorem 5. For every Borel-measurable function f the operator
Z
P (f ) =
f (λ)dP (λ),
D(P (f )) = Df
(129)
R
is normal and satisfies
||P (f )ψ||2 =
Z
|f (λ)|2 dµψ (λ),
Z
hψ|P (f )ψi =
R
f (λ)dµψ (λ)
(130)
R
for every ψ ∈ Df .
For bounded f complex conjugation, multiplication, and addition of functions can be mapped onto
the operators as follows in more or less obvious ways:
1. P (f )∗ = P (f ∗ )
2. Linearity, with domains adjusted to D(αf + βg) = D|f |+|g|
3. Product, with domain adjusted as D(f g) = Dg ∩ Df g
Proof. We first show that P (f )∗ = P (f ∗ ): We start from bounded functions fn = f χΩn , clearly
D(P (f )) = Df = Df ∗ and
hφ|P (fn )ψi = hP (fn∗ )φ|ψi
for φ, ψ ∈ Df = Df ∗ , which extends to the limit n → ∞ by continuity. In particular φ ∈ Df ∗ ⇒ φ ∈
D(P (f )∗ ) or D(P (f )∗ ) ⊇ Df .
Remains to show D(P (f )∗ ) ⊆ Df . The property P (fn ) = P (f )P (Ωn ) follows from the construction
of P (f ). With ψ ∈ D(P (f )∗ ) and any φ ∈ H we have
hP (fn∗ )ψ|φi = hψ|P (fn )φi = hψ|P (f ) P (Ωn )φ i = hP (Ωn )P (f )∗ ψ|φi.
| {z }
(131)
∈D(P (f ))
As we know that P (f ∗ )ψ ∈ H for ψ ∈ D(P (f )∗ , we can write
Z
Z
||P (f ∗ )ψ|| = lim ||P (Ωn )P (f ∗ )ψ|| = lim
χΩn |f ∗ |2 dµψ = |f |2 dµψ ,
n→∞
n→∞
i.e. ψ ∈ Df .
Normality trivially follows by inserting ||P (f )∗ ψ|| = ||P (f ∗ )ψ|| = ||P (f )ψ||.
52
(132)
10.3.1
The subspace Hψ
We now repeat the construction used for Hilbert space representations of a C ∗ algebra using the resolution
of identity and the map P (f ), but taken the Hilbert space as given and defining the state as ωψ (A) =
hψ|Aψi.
Definition 9. We denote by Hψ ⊆ H the closed subspace
Hψ = {P (g)ψ|g ∈ L2 (R, dµψ )}
(133)
Hψ is a closed subspace, as the L2 is closed and a sequence P (gn ) converges if and only if gn converges.
As P : g → P (g)ψ conserves the norm, ||g||L2 = ||P (g)ψ||Hψ , the existence of a limit in L2 implies that
P (g)ψ ∈ Hψ and vice versa.
We recognize the construction similar to the one used during GNS and expect that Hψ is in some
sense an invariant (not necessarily irreducible) subspace under the action of P (f ). However, as we admit
unbounded functions now, we need to deal with domain questions again and we cannot as easily reason
by purely algebraic arguments. Instead, we introduce the following:
Lemma 6. The subspace Hψ =: Pψ H. reduces the operator P (f ); i.e. Pψ P (f ) ⊆ P (f )Pψ .
Remember: A ⊆ B means D(A) ⊆ D(B), Af = Bf ∀f ∈ D(A). This expresses that the operators P (f )
“commute” with the projector Pψ onto the subspace Hψ at least for functions from Df . It will allow us
to define the operator restriced to the subspace Pψi P (f )Pψi , as in the unbounded case.
Proof. For f bounded it is easy, as we can use commutativity of multiplication without worrying about
domains. We study the action of the operators on any φ by decomposing it into its part in Hψ and the
orthogonal complement. Let φ ∈ Df = H, then there exists g ∈ L2 (R, dµψ ) such that φ = P (g)ψ + φ⊥ .
Clearly, for any h ∈ L2 (dµψ ) we have hP (h)ψ|P (f )φ⊥ i = hP (f ∗ h)ψ|φ⊥ i = 0 as f ∗ h ∈ L2 . As this holds
⊥
for all h, we know P (f )φ⊥ ∈ Hψ
.
Pψ P (f )φ = Pψ P (f )(P (g)ψ + φ⊥ ) = Pψ P (f g)ψ = P (f g)ψ = P (f )P (g)ψ
| {z }
(134)
∈Hψ
For unbounded, we first use the standard trick of restricting to functions fn from the domain converging to f . For every such bounded function we can reason as above and we find for φ ∈ Df
P (fn )Pψ φ = P (f )P (Ωn )Pψ φ → Pψ P (f )φ.
(135)
However, it is not clear whether P (f )Pψ φ ∈ Df , even if φ ∈ Df . After all, we are cutting from the
function φ only the piece that is in Hψ . Here we use that P (f ) is normal and therefore closed: as
P (f )φn = P (f )P (Ωn )φ is convergent, the limit φn → φ ∈ D(P (f )).
By that we can decompose any operator P (f ) into Pψ P (f )Pψ ⊕ P (f )⊥ : because of the projector
⊥
properties and the “commutation” there are no “cross-terms”: φ ∈ Hψ and χ ∈ Hψ
0 = hχ|Pψ P (f )φi = hχ|P (f )Pψ φi = hχ|P (f )φi
10.3.2
(136)
Unitarity, cyclic vectors, spectral basis
The map between functions g ∈ L2 (R, dµψ ) and function P (g)ψ ∈ Hψ is unitary, i.e. linear, bijective
by construction and it conserves the norm. We call ψ cyclic if Hψ = H. For cyclic vectors, we have
gained a representation of an arbitrary (possibly unbounded) operator P (f ) on Hψ = H, that is unitarily
equivalent to a representation of P (f ) as multiplication on L2 (R, dµψ ). For non-cyclic vectors, we iterate
the procedure as in the C ∗ algebra case until the (countably many) vectors of H are exhausted. A
sequence of such vectors ψi is called spectral basis
All in all, given an algebra of now unbounded functions on R, we have re-established properties as we
had seen it in the bounded case. What is left is to find the projection valued measure that will correspond
to the possibly unbounded self-adjoint operator A and a given ψ.
53
10.4
For every projection valued measure there is a self-adjoint operator
We choose for a(λ) = λ the operator P (a) =: A (with domain Da ) is self-adjoint: it is manifestly
hermitian and we have shown that P (a) is normal, i.e. D(A) = D(A∗ ). Also, for this particular operator,
the construction above gives its “spectral representation” in the proper sense, i.e. on L2 (dµψ ) it acts as
multiplication by its spectral values.
10.5
For every self-adjoint operator there is a spectral measure
We are almost done. We have seen that, if there is a resolution of identity P (λ), we can obtain operators
in H and, for the operator P (a) we already have its spectral representation. The last step is to find
a spectral measure dµψ and the corresponding resolution of identity such that the operator acts as the
multiplication by λ for the Hilbert space over R with that measure: L2 (R, dµψ ). We will appeal here to
a few facts from measure theory.
10.5.1
The resolvent of P (a)
Let a(λ) = λ be the identical function and denote A = P (a). For complex z, the operator
Z
RA (z) = (λ − z)−1 dP (λ)
(137)
R
is the resolvent of A. The corresponding quadratic form
Z
Fψ (z) = (λ − z)−1 dµψ (λ)
(138)
R
is called the Borel transform of the measure µψ . This function clearly maps the upper half plane onto
itself and is analytic on the upper half plane.
10.5.2
A measure is uniquely defined by its Borel transform
Interestingly, there is a one-to-one relation of functions as above and measures on R:
Theorem 6. Let the function Fψ (z) be any function that is holomorphic in z and maps the upper halfplane onto itself (Herglotz). Then it is the Borel transform of a uniquely defined measure µψ :
Z
1
dµψ (λ)
(139)
Fψ (z) =
R λ−z
The measure µψ is finite
Z
µψ < ∞.
It can be explicitly constructed by the Stieltjes inversion formula.
Z
1 λ+δ
µψ (λ) = lim lim
=(Fψ (t + i))dt
δ↓0 ↓0 π −∞
(140)
Remark Me make no attempt on proving this. We only point out that that limit is frequently encountered in the form
1
1
lim =
= δ(x − x0 ).
(141)
↓0 π
x − x0 + i
The limit δ ↓ 0 just makes sure that we approach the final projectors in a continuous way, as we only
have right-continuity.
Problem 10.39: Stieltjes inversion formula
54
(a) Show that
lim
↓0
1
=
π
1
x − x0 + i
= δ(x − x0 ).
(142)
R
Hint: Recall the meaning of dxf (x)δ(x − x0 ) and compute the above limit. Use the residual
theorem for an integration path along a circle around x0 .
(b) Show, formally, the Stieltjes inversion formula
µψ (λ) = lim lim
δ↓0 ↓0
Z
1
π
λ+δ
=(Fψ (t + i))dt,
(143)
−∞
where Fψ is the Borel transform of µψ :
Z
Fψ (z) =
(λ − z)−1 dµψ (λ)
R
10.5.3
The resolvent defines a measure dµψ
By the theorem above, we obtain a spectral measure µψ , that we had constructed more directly in the
case of bounded operators.
Lemma 7. For a self-adjoint operator A the function
Fψ,A (z) = hψ|(A − z)−1 ψi.
(144)
is Herglotz.
Problem 10.40: Herglotz property Let A be self adjoint and
Fψ,A (z) := hψ|(A − z)−1 ψi.
(a) Show that Fψ (z) is analytic in a vicinity of any point z0 with =m (z0 ) 6= 0.
Hint: Expand (A − z)−1 around z0 using the resolvent formula.
(b) Show that for =m (z) > 0 also =m (Fψ (z)) > 0
Note: We had shown that for self-adjoint operators σ(A) ⊆ R. This is equivalent to saying that the
resolvent of A
RA (ψ) = (A − z)−1 ψ
(145)
exists as a bounded operator for =(z) 6= 0 and is defined on all of H. Now consider the function
Fψ,A (z) = hψ|(A − z)−1 ψi.
(146)
This function is analytic in the upper half plane: one can show this using the resolvent formula for z in
a sufficiently small surrounding of z0 . Because self-adjointness of A, it maps the upper half-plane onto
itself: let z = x + iy and φ ∈ D(A) : (A − z)φ = ψ
=(hψ|(A − z)−1 ψi) = =(h(A − z)φ|φi) = h−iyφ|φi = iyhφ|φi.
(147)
Therefor there exists a measure µψ such that this function is its Borel-transform. Therefore, by the
Stieltjes inversion we obtain a spectral measure µψ .
Note: Analyticity of matrix elements of the resolvent is a very useful property: it allows us to, for
example, change contours in the integration and, in the end, defines the resolvents themselves as being
analyic in z.
55
10.5.4
A resolution of identity P (λ)
We have stated earlier (without proof) that a semi-bounded and closed quadratic form corresponds to a
self-adjoint operator. Indeed, the functional
Z
q(−∞,λ] (ψ) =:
χ(−∞,λ] (t)dµψ (t)
(148)
R
is non-negative (i.e. semi-bounded). Closedness
Z
λ
Z
∞
dµψ ≤
qλ (ψ) =
−∞
dµψ (λ) < ∞
(149)
−∞
Therefore an operator P (λ) exists, whose quadratic form is qλ .
Next one shows that the P (λ) form a resolution of identity; here, as we are dealing with bounded
functions throughout, no substantially new questions arise and we skip the details.
10.5.5
The spectral projectors
Lemma 8. The family of operators PA (Ω) defined by
Z
hφ|PA (Ω)χi =
χΩ (λ)dµφ,χ (λ)
(150)
R
forms a projection valued measure.
10.5.6
Spectral theorem for self-adjoint operators
To every self-adjoint operator A there corresponds a unique projection valued measure PA such that
Z
A=
λdPA (λ)
(151)
R
This theorem is central to all we do in QM. Among others, it allows us to define functions of a s.a.
operator as
Z
f (A) =
f (λ)dPA (λ).
(152)
R
This definition of functions f (A) has the virtue of clearly settling the domain questions and being applicable a to a wide class of functions, including discontinuous functions like the characteristic function.
Even constructions like “δ(A − a)” can be given meaning and we will used them in scattering theory.
11
Stone’s theorem
It was mentioned that there is a one-to-one relation between self-adjoint operators and unitary groups like
the time-evolution or continuous symmetry transformations. The latter have the much more immediate
physical interpretation and justify why we insist to working with self-adjoint “observables” in quantum
mechanics.
11.1
A self-adjoint operator generates a unitary group
Theorem 7. Let A be self-adjoint and let U (t) = exp(−itA) Then
1. U (t) is a strongly continuous one-parameter unitary group.
56
2. The limit limt→0
1
t
[U (t)ψ − ψ] exists if and only if ψ ∈ D(A) and it holds:
lim
t→0
1
[U (t)ψ − ψ] = −iAψ
t
(153)
3. The domain is invariant under U (t) : U (t)D(A) = D(A).
A one-parameter unitary group is a family of unitary operators U (t1 + t2 ) = U ((t1 )U (t2 ), U (0) = 1
s
(from which U (t)−1 = U (−t) = U ∗ (t)). Strongly continuous means that for tn → t, U (tn ) → U (t).
Proof.
• 1, group property: the standard exponential has this property, so in the spectral representation we can read off the group property
• 1, strong continuity:
Z
lim ||e−itn A ψ − e−itA ψ||2 = lim
|e−itn λ − e−itλ |2 dµψ (λ)
tn →t
tn →t R
Z
=
lim |e−itn λ − e−itλ |2 dµψ (λ) = 0.
R tn →t
We can exchange the limit with the integral as we have dominated convergence, i.e. point-wise
convergence and domination by an integrable function, viz. the constant function 4 ≥ |e−itλ −e−itλ |2 .
• 2, for ψ ∈ D(A) the limit exists
1
lim ||e−itn A ψ − e−itA ψ + iAψ||2 = lim | (e−itλ − 1) + iλ|2 dµψ (λ) = 0
tn →0 t
tn →0
(154)
• 2, if the limit exists, ψ in the domain: we will show that any operator where the limit exists, is
symmetric. Then, as a self-adjoint operator is maximal, any such operator coincides with A. Denote
by à an extension of A on the domain of all functions where a limit à exists, then à is symmetric:
i
−i
hφ|Ãψi = lim hφ| [U (tn ) − 1] ψi = lim h
[U (−tn ) − 1] φ|ψi = hÃφ|ψi
tn →t
tn →t t
t
(155)
But self-adjoint operators are “maximal”, i.e. Ã = A.
• 3: invariance: a simple consequence of the commutative group property: if the limit exists for ψ,
i.e. ψ ∈ D(A), then it also exists at U (s)ψ
1
1
1
|| U (t) − 1 ψ|| = ||U (s) U (t) − 1 ψ|| = || U (t) − 1 U (s)ψ||
(156)
t
t
t
One can also show that this is the only solution to the Schrödinger equation.
11.2
Stone theorem - unitary groups derive from self-adjoint operators
This inversion of the above is physically more interesting: we expect that the time-evolution should be
continuous in time and that it conserves the norm. It is not completely obvious why it would be linear.
The latter one can argue if we want the time-evolution that cannot lead to a decrease of von Neumann
entropy. This is plausible, as the simple time-evolution should not change the probability of a given
statistical mixture. If we accept the premises above, the Schrödinger equation is a consequence.
Theorem 8. (Stone) Let U (t) be a weakly continuous one-parameter unitary group. Then it has the
form U (t) = exp(−itA) for some self-adjoint A.
57
We can here use “weaker” conditions on U (t): as the U (t) are bounded, weak continuity implies strong
continuity. (Proof can be found in the literature, we have not discussed some of the rather basic tools to
do that proof.)
Proof.
• A is densely defined. For now, we do not know any domain, and picking any ψH could make
the limit fail. We construct a certain “fuzzyness” w.r.t. to the time-evolution into our state, such
that we can be certain that the limit exists.
• Take any ψ ∈ H and define
Z
τ
ψτ :=
U (t)ψdt
(157)
0
(the integral is well-defined as U (t) are bounded), which implies
are dense.
1
τ ψτ
converges to ψ, i.e. the ψτ
• We can do the limit for the ψτ : the special construction allows us to interchange the roles of U (t)
with U (τ ), such that the limit τ → 0 can be taken:
1
[U (t)ψτ − ψτ ]
t
Z
Z
1 t+τ
1 τ
=
U (s)ψds −
U (s)ψds
t t
t 0
Z t+τ
Z t
Z τ 1
... −
... −
...
=
t 0
0
0
Z
Z
1 t
1 t+τ
U (s)ψds −
U (s)ψds
=
t τ
t 0
Z
Z
1 t
1 t
t→0
U (s)ψds −
U (s)ψds −→ U (τ )ψ − ψ.
= U (τ )
t 0
t 0
(158)
(159)
(160)
(161)
(162)
The dense set of ψτ is contained in D(A). Clearly, the limit must be symmetric and the domain is
invariant under the action of the group, as before.
• ker(A∗ − z ∗ ) = {0} for =(z) 6= 0: suppose this were different, i.e. there is an eigenfunction
A∗ φ = z ∗ φ. Then for ψ ∈ D(A) we can construct an exponentially growing matrix element:
d
hφ, U (t)ψi = hφ, −iAU (t)ψi = −ihA∗ φ, U (t)ψi = −izhφ|U (t)ψi
dt
(163)
hφ|U (t)ψi = exp(−izt)hφ|ψi. But the lhs is bounded, therefore the only choice is hφ|ψi = 0 for all
ψ ∈ D(A).
• Need to skip: the fact that the adjoint of a symmetric operator has complex eigenvalues is the
typical indication that either we cannot extend the symmetric operator to a self-adjoint operator
or that the extension is not unique. Conversely, if there are no such eigenvectors, the symmetric
A is “essentially self-adjoint”, i.e. its closure A = A∗∗ is self adjoint and this extension is unique.
(Without proof: if you are looking for the proof, check for “defect indices” or “Caley transform”).
• We can now compare U (t) with V (t) = exp(−itA). We denote a possible difference as ψ(t) =
[U (t) − V (t)]ψ for ψ ∈ D(A). One calculates
lim
s→0
ψ(t + s) − ψ(t)
= iAψ(t) :
s
(164)
We find the norm of ψ(t) does not change in time
d
||ψ(t)||2 = 2<hψ(t), iAψ(t)i = 0,
dt
58
(165)
and as at time t = 0 there is no difference, there is never a difference, i.e the time-evolutions agree
on D(A), and as U (t) is bounded, the extension to V (t) is unique.
Problem 11.41: Weak and strong continuity Weak convergence of a sequence ψn * ψ
means
hχ|ψn i → hχ|ψi
∀χ ∈ H.
(166)
(a) Let ψn * ψ. Show that ψn → ψ if lim ||ψn || = ||ψ||.
(b) Let B(t) a family of unitary operators that is weakly continuous at t = 0, i.e.
lim hφ|B(t)χi = hφ|B(0)χi
t→0
∀φ, χ ∈ H.
(167)
Show that this implies strong continuity at t = 0: limt→0 ||B(t)χ|| = ||B(0)χ||
Problem 11.42: Stone theorem Complete the proof of Stone’s theorem. We had constructed a
domain where the limit limt→0 (U (t) − 1)/t = A is essentially self-adjoint. Denote by V (t) = exp(−itA),
where A is the closure of A. Show that U (t) = V (t).
Hint: Show that the evolutions U (t) and V (t) agree for ψ ∈
cD(A) and remember B.L.T.
12
12.1
Further important topics
Self-adjoint Hamiltonians
We know that −∆ is self-adjoint, what happens if we add a potential? One finds that if a s.a. operator A
is modified by a symmetric operator B that is not “too big” compared to A the result is again selfadjoint.
This the content of the following
Theorem 9. (Kato-Rellich) Let A be selfadjoint and B with D(B) symmetric. If there are constants
0 ≤ α < 1 and 0 ≤ β such that
||bψ|| ≤ α||Aψ|| + β||ψ||
∀ψ ∈ D(A),
(168)
then A + B, D(A + B) = D(A) is self-adjoint. If A is essentially selfadjoint, then A + B is essentially
self-adjoint with A + B = A + B.
The proof is simple IF we use the same fact that we used in the proof of Stone: (A ± iη)D(A) = H
for some η ⇔ A selfadjoint, i.e. neither half-complex plane is in the spectrum of A∗ .
Proof. First we find a bound for the norm of the resolvent ||(A ± iη)−1 || ≤ |η|−1 we had done that
before (e.g. problems09), it is easy to see in the spectra representation of the selfadjoint A and clearly
||A(A ± iη)|| < 1. Then we write
(A + B ± iη)D(A) = [1 + B(A ± iη)−1 ] (A ± iη)D(A) .
|
{z
}|
{z
}
I
(169)
II
The space II is = H (or dense), as A is selfadjoint (or essentially s.a.), the first part has ker(I) = {0} as
B(A ± iη) < 1 by the relative boundedness of B: with ψ = (A − iη)φ, ||ψ|| = ||(A − iη)φ||
||B(A ± iη)−1 ψ|| =≤ α||Aφ|| + β||φ|| < α||A(A ± iη)−1 ψ|| + βη −1 ||ψ|| < ψ
for sufficiently large η. Therefor ker(1 − B(A ± iη)−1 ) = {0}.
59
(170)
12.2
Self-adjointness of typical Hamiltonians in Physics
We know that the Laplacian can be defined as a self-adjoint operator. We know for sure in the Fourier
space, where the Fourier transform F(−∆) ∝ ~k 2 is a multiplication operator. The ~k 2 is a continuous
functions therefore measurable and with the domain
Z
D(~k 2 ) = {φ(k)| d(3) k|~k|4 |φk |2 < ∞
(171)
it is self-adjoint.
The Fourier-transform is defined on all L2 (R3 , dr(3) ) and invertible (even unitary, i.e. norm-conserving),
therefore the Laplacian is s.a. on the domain
D(−∆) = {χ|χ = F(φ), φ ∈ D(~k 2 )}.
(172)
That rather obvious fact can also be proven by considering that −∆ is quite obviously symmetric and
using our criterion of self adjointness (A ± i)D(A) = H:
F −1 (∆ ± i)D(∆) = F −1 (∆ ± i)FF −1 D(−∆) = (~k 2 ± i)D(~k 2 ) = H,
where the last equality follows from ~k 2 being s.a.
12.2.1
Domain of essential self-adjointness
The domain D(∆) is not very accessible in practice. Here the concept of essential self-ajointness plays
out its use: we can rather easily define a domain De (∆) essential self adjointness
De (∆) = Cc∞ (R3 ),
(173)
i.e. the infinitely differentiable functions with compact support. The closure of this is the self-adjoint
Laplacian as defined above:
(∆, De ) = (∆, D).
(174)
We leave this without proof.
12.2.2
Self-adjoint −∆ + V
For a large class of potentials the Hamiltonians on L2 (d(3) r, R3 ) are self adjoint. These are all potentials
of the form
V = w + v,
w ∈ L2 (R3 ), v ∈ L∞ .
(175)
This can be proven directly using Kato-Rellich with the help of estimates that are typical tools of
functional analysis. We will use a different approach that has further reaching implications. In the
proof, we have seen that it is important to estabilish invertibility of 1 + B(A ± iη)−1 to obtain all
H = (−∆ + V ± iη)D(∆). One has the feeling that, if only B(A ± i) is bounded, one can always increase
η until ||B(A ± iη)−1 || < 1. This is captured in the
Definition 10. A hermitian operator B ⊇ A is called relatively bounded to A if the map B : DΓ (B) → H
is continous w.r.t. the the graph norm of || · ||A .
A more transparently, one formulates this as
Corrolary 1. B is bounded relative to A iff there exists M such that
||Bφ|| ≤ M (||H0 φ|| + ||ψ||)
60
∀φ ∈ D(A).
(176)
We will prove this property for the Coulomb potential. Connecting to Kato-Rellich will be left as an
exercise.
Problem 12.43: Relatively bounded Show that the following three properties are equivalent
for s.a. operators A and B : D(B) ⊇ D(A):
1. B : DΓ (B) → H is continous w.r.t. the the graph norm of || · ||A .
2. There exists M such that
||Bφ|| ≤ M (||Aφ|| + ||ψ||) ∀φ ∈ D(A).
(177)
3. The operator B(A − z)−1 is bounded for at least one z.
12.3
Perturbation theory
Perturbation theory is based on the idea that if we add to an “unperturbed” self-adjoint H0 a perturbation
αH1 that we can make sufficiently small by α → 0, the spectrum and eigenvectors will depend analytically
on α such that an expansion for small α will approach the exact result. The problem is that, in infinite
dimensional Hilbert spaces, the idea of what is “small” needs to be made precise. For example, the dipole
interaction E~ · ~r of an atom with an electric field is small for some states (the ones localized close to zero),
but is arbitrarily large for states at a large distance. Conversely, the Coulomb potential is very large
at the origin, but rather weak at large distances. The key is, as above for Kato-Rellich, to look at the
strength of H1 relative to H0 .
12.4
Analyticity of eigenvalues
We study the analyticity of the resolvent Rz (α) := (H0 + αH1 − z)−1 as function of α. We can right
away write the formal power series
X k
(H0 − z + αH1 )−1 = (H0 − z)−1
αk H1 (H0 − z)−1 .
(178)
k
The rhs will norm-converge for all sufficiently small α, if H1 (H0 − z)−1 is bounded. The above proofs that
Lemma 9. Let H1 be relatively bounded w.r.t. H0 , i.e. H1 (H0 − z)−1 is bounded for =m (z) 6= 0. Then
the resolvent (H0 +αH1 −z)−1 is analytic w.r.t. to α and z in an open set of C×C including {0} ×ρ(H0 ).
12.4.1
Riesz projections
It is very useful to consider the following representation of projectors onto spectral subspaces
Lemma 10. Let E0 an isolated eigenvalue of a selfadjoint H and γ(E0 ) ⊂ ρ(H) a counterclockwise
closed path in the resolvent set that encloses E0 and but no other spectral values. The operator
Z
1
−
dz(H − z)−1 = PE0
(179)
2πi γ(E)
projects onto the subspace of eigenvectors to the eigenvalue E0 .
Note: by “isolated” we mean that there is such a γ(E0 ). The proof using the residual theorem is
trivial if we invoke the spectral theorem.
Problem 12.44: Riesz projection Define an eigenvalue with finite multiplicity as a spectral
value E0 ⊂ σ(H) of a s.a. H such the associated projector valued measure P (E0 ) is a finite-dimensional
61
projector. We call E0 isolated if there exists and open set U such that U ∩ σ(H) = {E0 }. Show the
properties of the Riesz-projections.
This is rather useful as it shows the intimate relation between spectral projectors (and with it the
spectral theorem) and the resolvent. Resolvents are in general easier to work with than the original
operator as they are bounded operators and the domain conundrums can be avoided. Obviously, by
encircling more than just one eigenvalue, we obtain projections onto larger parts of the spectrum.
Analytic functions are known to “survive” integrations: let f (z, x) be functions that are analytic in a
surrounding U of z0 for all x ⊂ γ. Here γ(t) describes some integration path in the complex plane. For
simplicity, assume that c is compact and f (z, x) is bounded uniformly for all x ∈ γ. Then the integral
Z
F (z) = dtf (z, γ(t))
(180)
is analytic w.r.t. z.
So if we can show analyticity of (H0 + αH1 − z)−1 , we have analyticity of projectors and with this
of the eigenvalues. By analyticity of a bounded operator we mean that it has a norm convergent series
expansion in terms of bounded operators, i.e. it can be written as
X
P (α) =
αk P (k)
(181)
k
where all P (k) are bounded.
Theorem 10. Let H1 be bounded relative to H0 . Then the isolated eigenvalues of H = H0 + αH1 with
finite multiplicity and the projectors onto them are analytic w.r.t. α in an open surrounding of α = 0.
The main (not very big) complication for the proof is that both, the spectrum and the spectral
projectors “move” at the same time, i.e. both depend on α. This, by the way, is not only a mathematical,
but also the main practical complication, that makes higher order perturbative formulae so ugly. Strategy
is to first show that the spectral projectors are analytic. Next compute a unitary transformation to an
operator, where the vectors do NOT depend on α and show that the eigenvalues depend analytically on
α.
Proof.
• As the resolvents exist and are analytic as functions of z for any sufficiently small range of
α, we can consider
Z
1
dz(H0 + αH1 − z)−1 ,
(182)
P (α) = −
2πi γ
where γ now includes z such that the resolvents remain analytic for all |α| < α0 . Such an environment exists by the above lemma.
• We can now replace the integrand by its power series in α and z and perform the integrations w.r.t.
z, thus analyticity of P (α) is proven.
• Note that P (α) has the same finite rank for α in the range of analyticity: We know that it is a
projector. For projectors dim(P (α)H) = Tr(P (α)). By asumption, T r(P (0)) is finite. We know
that Tr is continuous w.r.t. to the norm and as an analytic function the norm ||P (α)|| is continuous
w.r.t. to α. As the rank is a discrete quantity it cannot change its value in a continuous way.
• For given α we constrain the operator to the subspace of the eigenvectors
Hc (α) = P (α)H(α)P (α).
(183)
This is obviously an operator of finite rank. We will show that it is unitarily equivalent to an
operator that acts on the fixed subspace P (0)H.
62
• For that we write
Pc (α) = W (α)Pc (0)W ∗ (α).
(184)
The W (α) are operators that map from the unperturbed eigenspace to the perturbed eigenspace.
Formally, such a W (α) : Pc (0)H → P (α)H is
−1/2
W (α) = Pc (α) [1 + P (0)(Pc (α) − P (0))P (0)]
Pc (0)
(185)
The inverse exists as Pc (α) − P (0) is arbitrarily small, for the same reason the square root exists
and W (α) is analytic. The property (184) can now be verified by algebraic manipulations.
• The map W ∗ (α) : Pc (α)H → Pc (0)H conserves the norm: suppose ψα from P (α)H, then
||ψα ||2 = hψa |P (α)|ψα i = hψa W (α)P (0) W ∗ (α)ψα || = ||W ∗ (α)ψα ||2
| {z }
∀ψα .
∈P (0)H
and as the dimensions of the spaces agree, W (α) is invertible. As a map between the finitedimesional spaces W (α) is unitary: isometry+invertible=unitary.
• Now
Hc (α) = W (α) P (0)W (α)∗ H(α)W (α)P (0) W ∗ (α) :
{z
}
|
(186)
H̃(α), on P (0)H
what we have gained is that we can investigage the finite-dimensional matrix H̃(α): all functions
involved are analytic, and for the finite-dimensional H̃(α) this implies that also the eigenvalues are
analytic functions of α. Because of unitarity of W (α) they are identical to the eigenvalues of Hc (α).
Problem 12.45: Perturbation theory
(a) Show that for the projectors onto the unperturbed and perturbed sub-spaces it holds
Pα = Wα P0 Wα∗ .
for sufficiently small α with
−1/2
Wα = Pα [1 + P0 (Pα − P0 )P0 ]
P0
Instructions: Multiply from the left by Wα∗ and show Wα∗ Wα = 1, similarly for the other side. For
−1
the
P algebra, use [. . .]P0 = P0 [. . .] and evaluate Pα P0 (P0 Pα P0 ) P0 Pα . by writing the projectors as
k |k, xihk, x|, x = 0, α. Also verify and use P0 [. . .]P0 = P
(b) Show that map on finite-dimensional spaces W (α) : P (α)H → P (0)H is unitary.
63
13
Scattering theory
Literature: Reed and Simon, Vol III: Scattering Theory; Teschl, Mathematical Methods in Quantum
Mechanics.
13.1
Scattering is our window into the microscopic world
For example Rutherford experiment, COLTRIMS (Cold Target Recoil Ion Momentum Spectroscopy),
and many more. One rare non-scattering microscopic method: Scanning tunneling microscope.
Detectors can register: direction of emission, kinetic energy of arriving particle, possibly its internal
state (spin, excitation state, mass), arrival times. Arrival times are often used to determine kinetic energy
from knowing time of emission and distance to the point of arrival (“time of flight” detector).
13.1.1
Cross section
Given an ensemble of incident sub-system/particles characterized by {~ki , si , i = 1, . . . , I} with momenta
~ki , internal states si (e.g. spin, constituents, excitation state etc., if any), and flux Fi (number of such
subsystems per time per surface), one measures the number of sub-systems/particles emitted per time
Ne into momenta and internal states {~qe , se , e = 1, . . . , E}. The cross section is the ratio
N
e
σ ~k1 , s1 , . . . ~kI , sI ; ~q1 , s1 , . . . ~qE , sE =
Fi
(187)
This is what we ultimately need to know for comparison with observation.
13.1.2
Potential scattering
All deeper mathematical discussion in these lectures will be limited to the case of a single particle that
scatters without change of its internal state from a fixed, immutable target that is represented by a
potential. I.e. 1 = I = E and si = se , such that the only relevant two arguments of the cross section are
the initial and final momenta ~ki and ~qe . Physicists call this “potential scattering”.
Further we will not treat a statistical ensemble of particles but rather a particle described by a single
wave function. This is a minor technical simplification useful at the present level.
13.2
Mathematical modeling of a (simple) scattering experiment
Prepare a particle in some “free” state, e.g. free wave packet
Z
~
Φ(~r, −t) = (2π)−3/2 d(3) keik·~r ψ(~k)
(188)
at some time −t. Prepare the packet such that no part of it is in the range of interaction.
Let that wave packet evolve according to the scattering time-evolution. Wait long enough, say, until
t, and analyze the wave packet outside the range of the interaction.
The fundamental mathematical issues are:
• What does “preparing the state” mean? Can we actually map any arbitrary free wave packet into
a scattering wave packet? Is that mapping unique? (Existence and uniqueness of scattering
solutions.)
• What is the fate of the scattering state? Can the system just “swallow” an incoming particle? Or
can we be certain to find the system again in a (possibly quite different) free wave packet at some
large time t? (“Asymptotic completeness”)
64
13.3
The Møller operators
Now assume that far back in time a given wave packet can escape the action of the potential. At that
time, free and interacting time evolution are indistinguishable for the wave packet. Now we can consider
the wave packet as an interacting one and evolve forward in time to the present by the full time evolution.
Under which circumstances can we find times large enough to do such a mapping? Do the following limits
exist:
lim U (t)U0 (−t) := Ω± ?
(189)
t→±∞
In which sense may the limits exist? Can it exist in an operator sense? Given ,
?∃ T : ||U (t)U0 (−t) − Ω± || < ,
∀±t>T
(190)
(“Uniformly” for all states): probably not, we only need to choose arbitrarily slow wave packets, i.e.
particles that move extremely slowly out of the interaction region at a speed v < L/T , where L is some
characteristic dimension of the interaction region. However, it may exists “pointwise” for any wave packet
||[U (t)U0 (−t) − Ω± ]|Φi|| → 0 for ± t → ∞
(191)
That is, the time-limit can exist in the sense of a strong limit.
13.3.1
Properties
Note that the Møller operators map a wave-packet composed of free electrons (plane waves, exp(i~k~r) into
(±)
a wave packet composed of scattering solutions Ψ~ (~r) with asymptotic momenta ~k. We will find that
k
the energies remain unchanged in the process (as it should be). We expect, in some sense,
~
(±)
k
Ψ~ (~r) = Ω± eik~r .
(192)
and
~k 2 ~
~k 2
1
~
~
(±)
eik~r and (H0 + V )Ψ~ = HΨ~k =
Ψ~
(193)
− ∆eik~r = H0 eik~r =
k
2
2
2 k
We say “in some sense”, as the involved waves are not in Hilbert space.
For the absolutely continuous spectrum we expect an intertwining property of the Ω± , i.e.
HΩ± = Ω± H0 .
13.4
(194)
The Lippmann-Schwinger (LS) equation
Frequently, the first form how a physics student is exposed to scattering theory is in the form of the
(±)
Lippmann-Schwinger equation for the Ψ~ :
k
(±)
k
~
|Ψ~ i = lim |eik~r i −
↓0
1
(±)
V |Ψ~ i.
k
H0 − (k 2 /2 ± ı)
(195)
This is an equation for “scattering eigenfunctions” (± for “in/out”)
(±)
k
H|Ψ~ i =
k 2 (±)
|Ψ i
2 ~k
(196)
with the Hamiltonian
H = H0 + V
(197)
H0 the “free” Hamiltonian, V = H − H0 the interaction (potential), and Φ. . . “scattering eigenfunction”
of H0
H0 Φ = EΦ
(198)
65
Usual choice
1
H0 = − ∆ (or : − ∆)
(199)
2
The quotation marks indicate that the “eigenfunctions” are not in the Hilbert space and therefor their
existence and meaning need to be related to the operators that are defined as linear operators in Hilbert
space.
In position space the Lippmann-Schwinger equation is an integral equation
Z
0
1
e∓ı|k||~r−~r |
~
Ψ(±) (~r) = eık·~r −
d(3) r0
V (~r0 )Ψ(±) (~r0 )
(200)
4π
|~r − ~r0 |
It is not at all obvious how this equation is related to the above description of a scattering experiment. How, for example, would we extract a cross section from Ψ(±) ? And why would it be related
to the time-dependent definition of scattering given above? And how do Ψ(±) fit into our Hilbert space
formulation of quantum mechanics? Later in this introduction we will by formal manipulations derive
the equation. Historically, of course, it derives from the well studied scattering of classical waves. Here,
for the conceptually clear definition of scattering theory we use the limit of standard time-dependent
quantum mechanics to large times, as one does effectively in an experiment, where detection occurs at
macroscopically large times and at macroscopically large distances after a microscopic scale reaction.
Problem 13.46: Inverse of the Møller operator In the derivation of the LippmannSchwinger equation from the definition of the Møller operators we have used abstract reasoning to interchange φ ↔ ψ (±) . Instead, directly show
2
−1
Ω−1
V]
± = [1 + (H0 − k ± i)
13.4.1
(201)
In- and Out-states
Physically, we easily can imagine “outgoing” waves, i.e. waves that from a given finite time onwards will
(in some sense) move away from the scattering center. Similarly “ingoing” waves up to some finite time,
always move towards the scattering center. The signs in the Lipmann-Schwinger equation refer to these
two physical situations, but it is not completely obvious how to cast that idea into a clear mathematical
concept. In the end, we will be able to do this by looking at ~x · ~k: states ψ(t) where these are positive
will be “outgoing”, if negative ingoing.
It is important to note that in- and out-going are concepts of the long time limit, or, putting it
differently, they are related to the behavior of the solution at large distances. At close range, the concept
is not meainingful. What seems to be outgoing at one point may revert direction under the influence of
some force and become ingoing. Also, there is no possibility to characterize in- our outgoing by spectral
properties. For stationary states, as they appear in the LS equation, the total flux is conserved, so clearly
the in- and out-going parts some will be equal. An unabigous way to think about that concept will be
introduced in the following.
13.5
Spectral properties and dynamical behavior
In physics we like to think of bound states - that never seriously leave some finite region, and unbound
states, that are certain to leave any finite surrounding of the system. We also have got accustomed to
associating these two types of states with discrete and continuous spectrum, respectively. Not much of
a justification is given for this ususually. We will first establish that fact rigorously. The main theorem
expressing this fact is called RAGE (for Ruelle, Amrein, Georgescu, Enß).
13.5.1
Spectral subspaces
Given the separation of the spectral measure into point, absolutly continous and singular spectrum
hφ|dP (λ)ψi = dµφ,ψ = dµp;φ,ψ + dµac;φ,ψ + dµsing;φ,ψ
66
(202)
and the fact that the three parts are mutually singular (Lebesges decomposition theorem) we find that
we can decompose the Hilbert space into mutually orthogonal spaces
H = Hp ⊕ Hac ⊕ Hsing =: Hp ⊕ Hc .
(203)
We have lumped singular and ac into “continuous” Hc , as these two part behave similar in many respects
and actually, usual Schrödinger operators have empty singular spectrum. We omit the comparatively
simple proof to this fact. We may also write
Hp = span(eigenvectors)
(204)
Hc = Hp⊥
(205)
and
Functions ψ ∈ Hac behave like wave packets, i.e.
lim he−itH ψ|φi = 0,
(206)
t→±∞
we can say: after long time, the wave packet becomes orthogonal to any fixed state.
For functions from the sing spectrum a similar decay only holds in the time-averaged sense. This kind
of spectrum is poorly suitable for scattering considerations, as we cannot exclude that the system returns
to its initial state, even if only for a short time. Note that Hamiltonians with sing spectrum are not
necessarily pathological cases: they may possibly be realized in quasi-crystals (i.e. with quasi-periodic
potentials). We will not consider them for our discussion of scattering theory.
For proving long-time decay of continuous packets, we first state
Theorem 11. (Wiener) Let µ be a finite complex Borel measure on R and let
Z
µ̂(t) =
e−iλt dµ(λ)
R
be its Fourier transform. Then the Cesáro time average of µ(t) has the limit
Z
X
1 T
lim
|µ̂(t)|2 dt =
|µ({λ})|2 ,
T →∞ T 0
λ∈R
where the sum on the right hand side is finite.
Note that the sum is over individual points, not any finite intervals: the theorem somehow tells us
that on average the Fourier transform of the square reduces to a the measure of a discrete set of points.
One thing we readily observe is that here the ac and sing parts of the measure cannot contribute as they
are zero on discrete points. Basically, as we will see in the proof, the averaging over t (“time”) in the
limit T → ∞ generates δ-like functions.
Proof. As our measure is finite and the functions are measurable, we can apply Fubini’s theorem to
evaluate
Z
Z Z Z
1 T
1 T
|µ̂(t)|2 dt =
e−i(x−y)t dµ(x)dµ∗ (y)dt
T 0
T 0 R R
Z Z
Z
1 T −i(x−y)t
=
e
dtdµ(x)dµ∗ (y)
R R T 0
The t-average converges point-wise to χ{0} (x − y) and it is bounded, such that (by “dominated convergence”) we have
Z Z
Z
X
∗
=
χ{0} (x − y)dµ(x)dµ (y) =
µ({y})dµ(y) =
|µ({y})|2
R
R
R
67
y∈R
Remember similar manipulations that you will have seen in your quantum mechnics course: actually,
only the long-time limit establishes sharp, well-defined energies and only bound states can realize these
in a physical system.
Problem 13.47: Spectral subspaces We define the projectors onto spectral subspaces of A
as P (xx) = χxx (A) with xx = p, sing, or ac, Hxx = P (xx) H and χxx the characteristic functions of the
mutually disjoint subsets Mxx .
(a) Reassure yourself of the decomposition of H = Hp ⊕ Hac ⊕ Hsing
(b) Show that Hxx are invariant under the time evolution e−itA .
(c) Show that Hxx reduces A.
13.5.2
Fourier transform of smooth functions (Riemann-Lebesgue)
We had discussed the absolutely continuous measures as those that are equivalent to the Lebesgue measure
in the sense dµac (x) = f (x)dx with an integrable function f . That means that for the absolutely
continuous part of the spectral measure we can take advantage of the following
Theorem 12. (Riemann-Lebesgue) If Lebesgue integral of |f | is finite, then the Fourier transform of f
satisfies
Z
fˆ(λ) :=
f (x) exp(−iλx) dx → 0 as |λ| → ∞.
R
We leave this important and useful theorem without proof. It makes precise our intuition that a highly
oscillatory integral (|z| → ∞) will eventually average to zero, if the integrand is otherwise “smooth”, i.e.
without non-integrable singularities.
13.5.3
Compact operators
We have discussed bounded operators, and often they behave just as operators on a finite dimensional
space. But they are still acting in an infinite dimensional space and, when acting as potentials, the
can induce infinitely many (if not infinitely large) changes. For example, adding the (relatively) bounded
potential to the Laplacian, (like, for example the Coulomb potential) we can create infinitely many bound
states. Or, even more trivially, adding λ1 to an operator shifts the complete spectrum by λ.
First, we define genuinely “finite” operators:
Definition 11. An operator of finite rank is an operator whose range is finite dimensional. The
dimension of the range is called the rank of the operator
rank(K) = dim ran(K).
The general form of a compac operator is
Kψ =
n
X
φj hχj |ψi,
(207)
j=1
where the φj are an orthonormal basis for ran(K). Operators that can be norm-approximated by finiterank operators are called compact operators: We call K compact if there exists a sequence of finite
rank operators Kn : Kn → K (in the sense of the operator norm).
In a situation as above, such operator can add only a finite number of bound states to the Laplacian
and it cannot shift the continous spectrum. (We ommit the proof at this point.)check
Finally, what will be important for us is not that the operator itself is compact (e.g. the Coulomb
potential is not), but that it is compact relative to some other operator (e.g. the Laplacian). The
construction is a for relative bounded:
68
Definition 12. An operator K is compact relative to a s.a. operator A if D(K) ⊇ D(A) and K(A − z)−1
is compact for some z 6∈ R.
Theorem 13. Let H0 be s.a. and H1 D(H1 ) ⊇ D(H0 ) relatively compact, i.e. the operator H1 (H0 − z)−1
is compact. Then
σess (H0 + H1 ) = σess (H0 ).
(208)
The essential spectrum σess is all but the isolated point spectrum. I.e. it is σac ∪ σsing plus possible
limit points of the point spectrum. For illustration, think of the Hydrogen atom, where 0 is a limit point
of the point spectrum.
A very useful property of compact operators is that they turn a weakly convergent sequence into a
strongly convergent one.
Lemma 11. Let K be compact, then
ψn * 0 ⇒ ||Kψn || → 0.
(209)
To understand why this happens, remember that the typically weakly convergent but not normconvergent sequence ψn was the one that escaped to ever new directions. A compact operator is near-zero
on all but a finite subspace, i.e. on all but a finite set of directions. So the ψn at one point will be turning
to directions where the compact operator just beats them down.
Problem 13.48: Compact operators Let K be a compact operator and φn * φ, ||φn || < c.
Prove that ||K(φ − φn )|| → 0
[HVZ Theorem (Hunziker, van Winter, Zhislin, atomic Hamiltonian): existence of a lower bound,
stability of matter.]
13.5.4
The RAGE theorem
Relative compactness allows us to single out a “finite” subspace of in the spectral range of an operator.
The following theorem tells us, that no wave packet from the continuous spectrum can ever be contained
in such a finite subspace in the long run.
Theorem 14. Let A be self-adjoint and K be relatively compact and denote by P (c) H = Hc and P (ac) H =
Hac the projectors onto the continuous and absolutely continuous spectra, respectively, then
1
lim
T →∞ T
Z
T
||Ke−iAt P (c) ψ||2 dt = 0,
0
lim ||Ke−iAt P (ac) ψ||2 = 0
T →∞
for every ψ ∈ D(A). If in addition K is bounded, then the result holds for every ψ ∈ H.
The theorem tells us: if we evolve a continuous wave packet it will leave the finite space where K
differs significantly from A. Note that the norm ||e−iAt P (c) ψ||2 is constant, but its evolves into ever new
directions. If ψ is build from the absolutely continuous spectrum, that holds strictly. If ψ may contain
singular spectrum, there may be moments in time where the norm “revives” to be sizeable, but these
periods will be short.
Proof. Denote ψ ∈ Hc or ∈ Hac .
• For any finite rank operator the property is a direct consequence of Wiener’s theorem: Remember
||Kψ(t)||2 = ||
n
X
φj hχj |ψ(t)i||2 =
j=0
n
X
j=0
(the φj were chosen to be orthonormal)
69
|hχj |ψ(t)i|2
• Denoting (in the spectral rep)
Z
µ̂(t) =
dλe
−iλt
Z
hφj |P (λ)ψi =
dλe−iλt dµφj ,ψ (λ)
we recognize the situation of Wiener’s theorem. As we have restricted ourselves to Hc , the pointmeasure is zero. For the ac spectrum, we can directly refer to Riemann-Lebesgue.
• As sum over the χj is finite, the statements are valid for the sum as well.
• For K compact, choose a sequence of finite rank operators Kn → K such that ||Kn − K|| < 1/n.
Then
2
2
1
||Kψ(t)||2 ≤ ||Kn ψ|| + ||ψ|| ≤ 2||Kn ψ||2 + ||ψ||2
n
n
So the theorem holds for K compact.
• For the general case, choose any ψ ∈ D(A). As P (c) A = P (c) A (more precisely: Hc reduces A,
ψ ∈ Hc if and only if ψ = (A − i)−1 φ and φ ∈ Hc . With
Ke−iAt P (c) ψ = Ke−iAt P (c) (A − i)−1 φ = K(A − i)−1 e−iAt P (c) φ
|
{z
}
compact
we are back to the previous situation.
• If K is bounded, we approximate D(A) 3 ψn → ψ ∈ H and obtain the estimate
||Ke−iAt ψ|| ≤ ||Ke−iAt ψn || + n ||K||.
This theorem allows us to formulate the following theorem, that tells us that time-evolution separates
the (at least on average) the Hilbert space into parts that leave any finite subspace and parts that will
remain contained in a finite subspace.
Theorem 15. (RAGE). Let A be self-adjoint. Suppose there is a sequence of relatively compact operators
Kn which converges strongly to identity. Then
Z T
Hc = {ψ ∈ H| lim lim
||Kn e−itA ψ||dt = 0}
n→∞ T →∞
0
Hp = {ψ ∈ H| lim sup ||(1 − Kn )e−itA ψ||dt = 0}
n→∞ t≥0
This theorem tells us that we can characterize Hc and Hp completely by the behavior of functions
under the long-time limit of quantum mechanical evolution: the first part says that the fact that wave
packets composed of the continuous spectrum will be “diluted” out of any “finite” subspace. “Finite”
means the space where K differs relevantly from A (relatively compact). In physics terms: if a wave
packet fully spreads (in the time-average), then it completely belongs to the continuous spectrum.
The second part tells us that if we can find a “finite” space (in the sense as above) where the wavepacket remains contained, the wave packet belongs purely to the continuous spectrum: spectral properties
and time-evolution behavior match one-to-one.
Proof. Denote ψ(t) = exp(−itA)ψ, start with first part:
• Let ψ ∈ Hc a function from the contiuous subspace, then For using the very similar the previous
theorem we need to get a square into the integral. We do this by Cauchy-Schwartz (with a second
function g(t) ≡ 1):
!2
Z
Z
1 R
1 T
2
||Kψ(t)||dt ≤
||Kψ(t)|| dt
→ 0.
T 0
T 0
70
• If ψ 6∈ Hc , i.e. ψ = ψc + ψp . If we we find that then for some large n that ||Kn ψp (t)|| ≥ > 0, we
have made our point.
• In fact, we can even claim
lim sup ||Kn ψp (t) − ψp (t)|| = 0.
n→∞ t≥0
P
• We write ψp (t) j aj (t)ψj for orthonormal eigenfunctions. (Remember the ψp (t) remains in Hp .
Truncate this after N terms. Now we have a fixed subspace and by assumption of the theorem the
Kn converge strongly, i.e. the statement holds for the truncated function.
• We need to refer to an auxiliary lemma (Teschel Lemma 1.14, essentially the Banach-Steinhaus/uniform
boundedness) to infer from strong convergence of the Kn uniform boundedness: ||Kn || < M . This
allows to extend the statement from the finite sum to the complete ψ.
• For the second equation, first consider ψ ∈ Hp . From what we have just shown, (1 − Kn )ψ(t) → 0
• Finally, assume Hp 3 ψ = ψp + ψc . We are done, if we find that ||(1 − Kn )ψc (t)|| does not tend
to 0 with n. We show, that we obtain a contradiction if it were to tend to 0 with n. The also the
integral tends to 0, i.e. for given < ||ψc || there is an n such that
1
T →∞ T
Z
1
T →∞ T
Z
T
||(1 − Kn )ψc (t)||dt
> lim
0
T
||ψc (t)|| − ||Kn ψc (t)|| dt
≥ lim
0
T
Z
1
≥ ||ψc (t)|| − lim
T →∞ T
||Kn ψc (t)||dt = ||ψc || > 0.
0
Problem 13.49: Corrolary of RAGE
Let A be self-adjoint, Kn a sequence of relatively compact operators converging strongly to the identity
(these are the same Kn as for RAGE theorem). For ψ in the Hilbert space H = Hp ⊕ Hc (Recall the
decomposition of the Hilbert space into the pure point and the continuous subspace) define
1
P ψ = lim lim
n→∞ T →∞ T
Z
1
n→∞ T →∞ T
Z
p
P c ψ = lim lim
T
eitA Kn e−itA ψdt,
(210)
eitA (Id − Kn )e−itA ψdt.
(211)
0
T
0
Prove, that P p projects onto Hp and P c onto Hc .
13.6
The Møller operators
Definition 13. Let H and H0 be two self-adjoint opertors. The wave operators (Møller operators) are
defined as
D(Ω± ) = {ψ ∈ H|∃ lim eitH e−itH0 ψ}
t→±∞
Ω± ψ = lim e
t→±∞
itH −itH0
e
ψ
We call D(Ω± ) the set of outgoing/ingoing states and ψ± the states with ougoing/ingoing asymptotics.
71
Lemma 12. The sets D(Ω± ) and ran(Ω± ) are closed and Ω± : D(Ω± ) → ran(Ω± ) is unitary.
Note that the roles of H and H0 can be interchanged, it is our choice which of the two time-evolutions
we consider primary. It is just the fact that with H0 = −∆ we know the time-evolution very well that
favors the conventional choice.
Proof.
• Clearly, ||Ω± ψ|| = || limt→±∞ eitH e−itH0 ψ|| = limt→±∞ ||eitH e−itH0 ψ|| = ||ψ||
• D(Ω± closed: let ψn ∈ D(Ω± ), Cauchy, then Ω± ψn is Cauchy and the limit exists, i.e. ψn → ψ ∈
D(Ω± ).
• ran(Ω± ) closed: we interchange the roles of H ↔ H0 to see that ran(Ω↔
± ) = D(Ω± ), therefore also
ran(Ω± ) is closed.
• By the above we also see invertibility of the Ω± .
Note that we use a slightly more general definition of unitary as norm-conserving bijection.
13.7
S-matrix
As a first step for connecting to experimental observables, we ask: what is the probability of finding a
system, that in the remote past was a free wave packet state χi , say with a narrow momentum wavepacket around a given ~k, in the free momentum wave packet χe around ~q after scattering. For that
we map the two wave free wave packets into the actual scattering solutions φi and φe by applying the
scattering operators Ω+ and Ω− , respectively. Note that we choose two different signs, as we want to
map the ingoing free wavepacket χi to the scattering wave packet φi = Ω− χi , such that they are equal
at the remote past, and conversely φe = Ω+ χe with mapping in the remote future.
By general quantum theory, our probability is
hϕi |ϕe ihϕe |ϕi i.
(212)
hϕe |ϕi i = hΩ+ χe |Ω− χi i = hχe |Ω∗+ Ω− χi i =: hχe |Sχi i
(213)
All we need to know is
The operator S := Ω∗+ Ω− = Ω−1
+ Ω− is traditionally called S-matrix and obviously contains all information
that we are interested in for a scattering experiment. Note that at this point of the discussion the
adjoint Ω∗+ is not well defined. Only after showing asymptotic completeness of the Ω± we can identify
P (H)(ac) H ≡ P (H0 )(ac) H and thus define the standard adjoint. The scattering matrix S is then a unitary
map between the ac spectral parts of the free time evolution S : Pac (H0 ) → Pac (H0 ).
Note that in this reasoning the relation to the cross section is still a bit murky, as we do not clearly
specify at which point of their evolution we compare the two scattering wave packets. In the limit of
infinitely narrow wave packets, this question disappears: in this limit, the particles become completely
delocalized and it does not matter, at which time we compare the wave packets, the particles are “always
everywhere”. We will not explicitly go through the rather tedious exercise of actually taking the limit.
13.7.1
Asymptotic completeness
We already discussed that Ω± will not in general be unitary as maps H → H. At finite times t the operator
U (t)U0 (−t) is unitary. We know that bound states (any vector from the point-spectrum) cannot appear
in the range of Ω± (those states, according to RAGE, would never get out of some finite range).
Although the Ω± will not be unitary as maps on H, they can still be bijections. We would like
them to be bijections between the space of all free states, i.e. all wave packets formed in Pac (H0 )H
and all scattering states, which we define as wave packets from Pac (H)H. Note that for H0 = −∆:
Pac (H0 )H = H. If the Ω± are such bijections, we say that scattering is asymptotically complete.
72
The physical meaning of asymptotic completeness is that scattering states and localized states do not
mix and that the dynamics of the unbound (ac) states of the the two systems are equivalent: localized
remains localized, scattering remains scattering. It just does not happen, that a wave packet goes in and
turns into a bound state or gets stuck in some other way. All that goes in comes out again. Neither
does it happen that a localized state decides to decay. Of course, in physics such processes do seem to
occur, they go by the name of “capture” or “decay” or so: looking more closely, in these processes either
another constituent goes out (e.g. a photon is emitted) or some long-lived intermediate state is formed
(a resonance), that will decay eventually.
As often, an example where the scattering operators exist, but are not asymptotically complete is
interesting, (see Reed and Simon III, counterexample to Theorem XI.32, p.70).
13.7.2
Scattering operators compare two dynamics
We see that the roles of H0 and H enter symmetrically: the scattering operators connect two unitarily
equivalent dynamical systems. Scattering theory compares one known dynamics to another one, maybe
less well known. From a mathematical point of view, it is irrelevant, which is which. Physically, of course,
we know everything about, say, −∆ and often rather little about −∆ + V .
But, for example, we know also a lot about the Coulomb problem (all eigenfunctions and generalized
scattering eigenfunctions) and we could just as well use the Coulomb Hamiltonian as our H0 = −∆ −
1/r and compare another dynamics, that only asymptotically behaves Coulomb-like. Of course, the ac
“eigenfunctions” of −∆ are just plane waves, which are much easier to handle than the hypergeometric
scattering functions of the Coulomb problem.
Unbound Coulomb dynamics itself is not unitarily equivalent to free motion. We will need to construct
a different dynamics to establish such an equivalence (Dollard Hamiltonian).
13.7.3
Intertwining property
Looking at
lim eitH e−itH0 e−isH0 = lim e−isH eitH e−itH0
t→±∞
(214)
t±∞
we see
e−itH Ω± = Ω± e−itH0
−itH0
(215)
−itH
Also, the domains are trivially invariant under e
and ran(Ω± ) are invariant under e
. As a
consequence, also their complements are invariant under the respective time evolutions: D(Ω± ) and
ran(Ω± ) reduce the operators e−itH0 and e−itH respectively. Taking the time derivative at t = 0 we find
the
Theorem 16. (Intertwining property) The subspaces D(Ω± ) and ran(Ω± ) reduce H0 and H, respectively,
and the operators on these spaces are unitarily equivalent:
Ωω H0 ψ = HΩ±
for ψ ∈ D(Ω± ) ∩ D(H0 ).
(216)
With the above we seem to have achieved our goal. However, the domains D(Ω± ) may not be all of
P (ac) H, they may even be trivially {0}.
First we single out a situation where we obtain results as desired (“Cook’s criterion”):
Lemma 13. (Cook) Suppose D(H0 ) ⊆ D(H). If
Z ∞
||(H − H0 ) exp(∓itH0 )ψ|| < ∞ for ψ ∈ D(H0 )
(217)
0
then ψ ∈ D(Ω± ), respectively. It then also holds that
Z ∞
||(Ω± − 1)ψ|| ≤
||(H − H0 ) exp(∓itH0 )ψ||dt.
0
73
(218)
Proof. We use the Dyson equation
Z
t
U (t − s)(H − H0 )U0 (s),
U0 (t) = U (t) + i
(219)
0
or
e
itH −itH0
e
Z
ψ =ψ+i
t
eisH (H − H0 )e−isH0 ψds ∀ψ ∈ D(H0 ),
(220)
0
and the norm of this is finite by the assumption of the lemma.
With this we can immediately single out a class of potentials where the Møller operators are defined
on all H
Theorem 17. Let H0 = −∆ and H = H0 + V where V ∈ L2 (R3 ). Then the Møller operators exist and
D(Ω± ) = H.
Proof.
• We need to show finiteness of
Z
||V ψ(s)|| =
|V (x)ψ(s, x)|2 dx
ψ(s) := eisH0 ψ.
R
Considering ψ like a multiplication operator applied to V , we see
||V ψ(s)|| ≤ ||V ||||ψ||∞ ≤
1
||ψ(0)||1 ||V ||, s > 0,
(4πs)3/2
The last inequality will be left for an exercise. This estimate is useful if ||ψ||1 < ∞. with that we
get the estimate for the integral
Z ∞
1
||V (ψ(s)|| <
||ψ(0)||1 ||V ||
3/2
(4π)
1
Therefore we know ψ ∈ D(H0 ) and ||ψ||1 < ∞ is ∈ D(Ω+ ). Same procedure for Ω− .
Problem 13.50: Dyson equation Assume that
i
d
ψ(t) = H0 (t)ψ(t),
dt
i
d
φ(t) = H(t)φ(t)
dt
(221)
define unitary time-evolutions ψ(t) = U0 (t, t0 )ψ(t0 ) and φ(t) = U (t, t0 )φ(t0 ). Convince yourself of the
identity (“Dyson equation”)
Z t
U (t, t0 ) = U0 (t, t0 ) − i
U0 (t, s)[H(s) − H0 (s)]U (s, t0 )ds,
t0
13.8
Asymptotic completeness - guide through the proof by Enss
We have found a case where the Møller operators are defined as we want on D(Ω± ) = P (ac) (H0 )H. What
we do not know yet is, whether the desired property ran(Ω± ) = P (ac) (H)H holds. This is the core of the
proof of asymptotic completeness. We will follow an elegant proof by Volker Enß[Comm. Math. Phys.
61, 285, (1978)]. First outline of the ideas and steps of the proof (following Teschl)
Watch out: recent mathematical physics uses different sign conventions for the scattering operators
than physicists. The rift in sign conventions goes right between Reed and Simon, and, for example,
Teschl. As we mostly rely on Teschl for the proof, we use this convention.
The Theorem states:
Theorem 18. Suppose V is short range. Then the Møller operators exist and are asymptotically complete.
74
13.8.1
Idea of the proof
The idea of Enß’s proof is clearly spelled out in his original paper: take any wave packet ψ ∈ Pac (H) to
a remote time in the future e−itH ψ = ψ(t). At that time, decompose it into parts that go out, i.e. parts
where position vector and momentum point to the same direction from the scattering potential (assumed
to be located near the origin) ~x · p~ > 0, and parts that go in ~x · p~ < 0.
As a first step, this decomposition can be used to show that the scattering operators exist on all
P (ac) (H0 )H = H via Cook’s criterion. Once we know the scattering operators exist, we also know that
(intertwining property)
Ω± Pac (H0 )H = Pac (H)Ω± H ⊂ Pac (H)H
(222)
As asymptotic completeness is equivalent to
Ω± Pac (H0 )H = Pac H,
(223)
Ω± Pac (H0 )H ⊃ Pac H,
(224)
one only needs to show
i.e. any scattering wave packet ψ ∈ Pac (H)H is also ψ ∈ Ω± Pac (H0 )H.
This second and decisive step is made as follows: at large positive times t, outgoing components
cannot interact in the future, as they will never get near to the potential, from which one concludes that
outgoing components remain in ran (Ω+ ). Ingoing components, on the other hand, at sufficiently large
positive times, due to the particular short range nature of the potential, are essentially restricted to a
finite-dimensional space. The fact that the corresponding map will be a compact operator will guide our
definition of what is “short range”. However, we will also have a more constructive definition about the
behavior of the potential at large r.
As the scattering wave-packet converges weakly to 0, at large times it will become orthogonal to the
near finite-dimensional ingoing components. From this one can conclude that the outgoing wave-packet
is φ(t → ∞) ∈ ran Ω+ . One remembers that ran Ω± is invariant under the time-evolution, so if ever
φ(t) ∈ ran Ω± for any time, it stays there for all times. The time-limit t → −∞ shows φ(t) ∈ ran Ω− .
For the reasoning to work one needs to be able to neglect the potential at one point compared to
the kinetic energy parts of the operator, otherwise the identity of “ingoing” and “outgoing” cannot be
established with sufficient accuracy. This is what restricts the proof (and the present formulation of
scattering theory as a whole), to short range potentials (see above and Teschl Eqs. (12.29) and (12.30)).
13.8.2
Exclusion of the sing spectrum
Asymptotic completeness requires some reasoning in terms of spectral theory to exclude complications
with the singular continuous spectrum. Here we are helped by the fact that we know that −∆ has purely
ac spectrum. If perturbations are added that are not too wild, we will only generate a few bound states,
maybe resonances, but no spectrum of sing character. However, we skip this discussion here and assume
there is no sing spectrum.
13.8.3
In- and outgoing spaces
But how to decompose the wave packet into some kind of in- and out parts without resorting to the
scattering operators themselves? The division into in- and outgoing parts of the ψ is achieved by constructing ~x · p~ as a self-adjoint operator projecting onto the positive and negative spectral subspaces.
D = 21 (~x · p~ + p~ · ~x), the self-adjoint generator of the (unitary) dilation group
Uλ Ψ(~x) = (eλ )3/2 Ψ(eλ ~x),
λ
Uλ =: eie
D
(225)
By Stone theorem, the generator D of a differentiable, one-parameter unitary group (Uλ ) is selfadjoint
(Teschl (10.9) and (12.18)). Spectral projectors are used to split the space into parts with positive (out)
and negative (in) dilation rate, i.e. P± := PD ((0, ±∞)).
75
Problem 13.51: Dilation group and its generator Coordinate stretching by an overall
factor e−λ is defined as
φ(~r) → φ(e−λ~r).
(226)
(a) Adjust the above idea such that that the stratching map Uλ φ is unitary.
(b) Show that the Uλ form a unitary, strongly continuous group.
(c) Show that on a suitable dense set ⊂ H the generator Uλ = exp(iλD) is given by
D=
13.8.4
i
~ +∇
~ · ~r)
(~r · ∇
2
Mellin transform
We also will need an instrument to convert D into a multiplication operator, i.e. a spectral representation
of D. This is the Mellin transform, which can be motivated by the following considerations: in polar
coordinates, the dilation stretches (or shrinks) all distances from the origin by a factor eλ , λ ∈ R. So
the radial coordinate is actually multiplied by a number as in eλ r. We can consider the logarithm of this
ρ = log(r), for which the stretch will appear as a shift by λ. So that means that D generates a shift when
considering its action on ρ. If we take the Fourier transform w.r.t. ρ, D will appear as a multiplication
operator. Designating ~r = rω we have the Mellin transform
M :LR
n
ψ(rω)
1
→ (Mψ)(λ, ω) = √
2π
Z
→ L2 (R, S n−1 )
(227)
r−iλ ψ(rω)rn/2−1 dr
(228)
∞
0
The Mellin transform (as a Fourier transform) is unitary and it converts the stretching transforms U (s)
and D to multiplications
M−1 U (s)M = e−iλs
M−1 DM = λ
As the measure on the space is ac, we immediately know that D has pure ac spectrum.
Next, we observe Ff (D) = f (−D)F: the dilation operator changes sign under Fourier transform.
This is to be expected, leaving domain questions aside, as FxF −1 FpF −1 = −px.
We know that there exists a domain such that D is s.a. One can show that D is essentially self-adjoint
on the Schwartz space, i.e. with D(D) = S(Rn ).
Problem 13.52: Dilation operator Show that D, D(D) = S(Rn ) is essentially self-adjoint.
Hint: Map into a the spectral representation and show ker(D∗ ) = {0}.
13.8.5
Schwartz space
We will be interested in functions that decay well (better than any inverse polynomial), but smoothly
enough such that their derivatives also decay rapidly. This is the Schwartz space of functions
S(Rn ) = {f ∈ C ∞ (Rn )| sup |xα (∂β f )(x)| < ∞, α, β ∈ Nn0 }.
(229)
x
For α and β we use a multi-index notation. These functions are dense in the corresponding Hilbert space.
76
13.8.6
Compact operators, weak convergence
Note the following: a weakly convergent series φn * φ is uniformly bounded supn ||φn || = M < ∞.
One might be worried that a sequence, while turning to ever new directions could also grow in norm
indefinitely. But this cannot happen for Banach spaces, as a consequence of the uniform boundedness
principle
Theorem 19. (Banach-Steinhaus) Let X be a Banach space and Y some normed vector space. Let Aα
be a family of bounded linear maps. Suppose ||Aα x|| < Cx is bounded for fixed x ∈ X. Then Aα ≤ C is
uniformly bounded.
Proofs for this very fundamental theorem can be found in literature, also Teschl. (Note that in one
of the problems we did not use it and rather require the sequence to be bounded.)
We will define short range potentials such that they generate only relatively compact differences
between the two dynamics H and H0 . The following equivalent characterizations of compact operators
will be useful there (Teschel 6.9):
Lemma 14. Let K be a linear map on H. The following statements are equivalent
1. K is compact
2. K ∗ is compact
s
3. An bounded linear maps on H, An → A implies An K → AK
4. ψn * ψ implies Kψn → Kψ.
5. ψn bounded implies Kψn has a norm-convergent subsequence.
Proof. We only show 1.,2.⇒ 3.:
• We consider An → 0 without loss of generality.
• As we have assumed An to be bounded and ||An ψ|| → 0, it is unformly bounded ||An || < M .
Therefore we can study the finite rank case
||An K|| ≤ ||An ||||K|| = ||An ||||KN || + ||An ||
•
13.8.7
Short range potentials
The discussion is restricted to H0 = −∆ and H = H0 + V . First requirement is that the potential V be
“relatively bounded” to −∆, i.e. ||V (−∆ + 1)−1 || < ∞. Note that, as σ(−∆) = R+ is postive, we can
use the a real offset instead of the complex z for the general self-adjoint case.
A relatively bounded perturbation V of the Laplacian −∆ is called short range if
Z ∞
||V (−∆ + 1)−1 1R3 \Br ||dr < ∞
(230)
0
where Br := {x ∈ R3 , |x| < r}. Relative boundedness guarantees that the integrand is always finite.
This is not as strong a constraint as it may seem: remember that also the Coulomb potential is relatively
bounded to −∆. What matters is that the remote parts of the potential drop sufficiently fast compared
to the kinetic energy. That this is not the case for the Coulomb potential.
77
Problem 13.53: Relative boundedness Show that the Coulumb potential is bounded relative
to −∆. Show that the Coulomb potential is not short range.
Technically, the short range property is mostly exploited in the form of the compactness of
(H − z)−1 − (H0 − z)−1
and f (H) − f (H0 )
(231)
Lemma 15. If V is short range, then RH (z) − RH0 (z) is compact.
For the proof of the lemma we would need a few results on compactness that we have skipped.
However, it gives some insight to consider the ideas of the proof
• Denote χr (x) = χ{x||x|>r} (x), the projector onto the remote parts of space
• So, 1 − χr projects onto a finite range.
• We use the (1 − χr )RH0 (z) is compact (without proof). In fact, the product of any two operators
g(x)f (p) for integrable functions g, f that deay to 0 at large x and p turns out to be compact (Teschl
7.21).
• Also, RH (z)V is bounded: that is plausible, as RH0 (z) is bounded and H = H0 + V , closely related
to Kato-Rellich. (Proof in Teschl 6.23).
• Now the short range condition enters
lim RH (z)V (1 − χr )RH0 (z) → RH (z)V RH0 (z)
r→∞
• Note the identity
RH (z)V RH0 (z) = (H0 + V − z)−1 V (H0 − z)−1 = (H0 + V − z)−1 − (H0 − z)−1
Actually, short range implies that for any function that decays at ∞, we have
Lemma 16. Suppose RH (z) − RH0 (z) is compact. Then so is f (H) − f (H0 ) for any f ∈ C∞ (R) and
lim ||[f (H) − f (H0 )]χr || = 0.
r→∞
(232)
Proof. Again the proof involves technical results that were skipped (Teschl 6.21). But once we know that
f (H) − f (H0 ) is compact, we know that χr excludes an arbitrarily large space, i.e. any χr φ goes strongly
to zero. Multiplying by a compact operator does not change that. As the family of maps is bounded and
goes to zeor strongly, it goes to zero uniformly, i.e. in the operator norm.
13.8.8
Escape of the wave packet - Perry’s estimate
Just how successfully the wave-packet escapes the influence of the potential is quantified by Perry’s
estimate (Teschl, Lemma 12.5). To specify this, one needs to battle a few technical complications: first
of all, very slow particles escape the potential very slowly. For any estimate, we need to have a least
an of velocity. For any we can formulate a sufficiently strong estimate. This still provides a dense
set of functions to work with. Secondly, one has a similar problem with the division into in and out:
when particles are very close to the origin or spiral very slowly away from the origin, the distinction may
not be sharp enough, although velocities may be high. Again, we need to make the estimate starting
from some finite dilation rate R by using the projector PD ((±R, ±∞)) rather than P± = PD ((0, ±∞)).
Perry’s estimate then is
78
Lemma 17. (Perry’s estimate) Suppose f ∈ C0∞ (R) with supp(f ) ⊂ (a2 , b2 ) for some a, b > 0. For every
R ∈ R, N ∈ N there exists a constant C such that
||χ{x||x|<2a|t|} e−itH0 f (H0 )PD ((±R, ±∞))|| ≤
C
, ±t ≥ 0.
1 + |t|N
(233)
Before proceding to the proof, let us look at the meaning of the pieces here:
• First of all, we project onto a subspace of the dilatations, setting a lower of bound R on the out-going
(or in-going) nature of the functions. In particular, R = 0 (not expansion at all under stretching)
and some open surrounding of it are excluded.
• The we make another spectral cut, this time w.r.t. kinetic energy by excluding too slow and, for
convenience, also too fast particles.
• Then the free time-evolution is applied.
• Finally, we project onto a sphere that grows at the “speed” of the slowest particle admitted in the
system
• This probablility decays with time faster than any power.
Clearly, this is the general behavior we expect if scattering is to make sense.
Proof.
• Let ψ ∈ S(Rn ). For the manipulations we we move partially into Fourier space, where f (H0 )
and the time-evolution have a simple form.
• Introduce
ψ(t, x) =e−itH0 f (H0 )PD ((R, ∞))ψ(x)
=hKt,x , FPD ((R, ∞))ψi
=hKt,x , PD ((−∞, −R))ψ̃i :
The scalar product is a convenient way of writing the back-Fourier transform to point x. We have
defined
2
1
Kt,x (p) =
ei(tp −px) f (p2 )∗
(2π n/2 )
i.e. we have lumped together the time-evolution in p-space, the Fourier transform and the kinetic
energy constraint, which is manifestly in L2 (d(n) p, Rn ). We can apply Schwartz inequality
hPD ((−∞, −R))Kt,x , ψ̃i2 ≤ ||PD ((−∞, −R))Kt,x ||2 ||ψ̃||2
which reduces the estimate to
||PD ((−∞, −R))Kt,x ||2 ≤
const
(1 + |t|)2N
for |x| < 2a|t|, t > 0
• For evaluating ||PD ((−∞, −R))K|| we use the Mellin transform (remember it is unitary)
||PD ((−∞, −R))Kt,x ||2 =
Z
−R
Z
dλ
d(n−1) ω|(MKt,x )(α, ω)|2
S n−1
−∞
where the integrand is
MKt,x )(α, ω) =
1
(2π)(n+1)/2
79
Z
0
∞f˜eiα(r)
• This integral has the form of a smooth function f˜ times a (rapidly) oscillating part (for details see
Teschl). For the case, the method of stationary phase provides a rigorous estimate
|MKt,x )(α, ω)| ≤
const
(1 + |λ| + |t|)N
which leads to the desired bound (for the missing details, consult Teschl).
Problem 13.54: Method of stationary phase Integrals of oscillatory functions are often
approximated in physics by this method. The rigorous mathematical formulation of the method comes
in the form of a bound as follows: Consider
Z ∞
If,φ (t) =
f (r)eitφ(r) dr,
−∞
where f ∈ Cc∞ (R) (i e. a compactly supported infinitely many times differentiable function R → C),
φ ∈ C ∞ (R) real-valued. Suppose now |φ0 (r)| ≥ 1 for all r ∈ supp(f ) (φ0 denotes the derivative). Show:
∀N ∈ N, there is a CN ∈ R, s t. |If,φ (t)| ≤ CN t−N .
Hint: Start with N = 0. For N > 0 use a change of variables to show that it is sufficient to look at
φ(r) = r. Then use partial integration.
Corrolary 2. Suppose that f ∈ C0∞ ((0, ∞)) and R ∈ R. The the operator PD ((±R, ±∞))f (H0 ) exp(−itH0 )
coverges strongly to 0 as t → ∓∞.
This states that, under suitable spectral cuts, at large postive times there remain no ingoing parts in
the any wave packet undergoing free time-evolution, at large negative times, there are no outgoing part.
For the simple proof check Teschl.
Problem 13.55: Corrolary to Perry’s estimate Recall Perry’s estimate: Let f ∈ Cc∞ (R)
with supp(f ) ⊂ (a2 , b2 ) for some a, b ∈ R. Then for every R ∈ R, N ∈ N there is C ∈ R, s. t.
kχ{x:|x|<2|at|} e−itH0 f (H0 )PD ((±R, ±∞))k ≤
C
,
(1 + |t|)N
where PD is the projection valued measure associated to the dilation operator D = (xp + px)/2.
Prove the following corollary: Let f as above and R ∈ R. Then
s
PD ((±R, ±∞))f (H0 )e−itH0 −−−−→ 0
t→±∞
(Recall, that strong convergence means convergence of every vector the sequence of operators is applied
to.)
Hint: First split ψ into χψ + (Id − χ)ψ for a useful χ. Second use Perry’s estimate for the part near the
origin and show that the other part can be neglected. Don’t forget, that we are dealing with bounded
operators!
13.8.9
Existence of scattering operators for short range potentials
The existence of scattering operators is shown by Cook’s criterion, where the boundedness of the integrals
is shown by splitting the integral into an inner region and an outer region with a time-dependent radius
separating the two regions. In the outer region, it is finite because of the short range assumption. In the
inner region Perry’s estimate can be administered for a dense set of functions of the form where both
spectral constraints discussed above are applied (Teschl 12.32). Somewhat loosely speaking: scattering
80
operators exists because the interacting wave packet leaves from near the potential sufficiently rapidly
(Perry estimate) and at larger distances not much happens (short range).
The Møller operators exist by Cook’s criterion, if
Z ∞
dt||V e±itH0 φ|| < ∞
(234)
t0
for all functions φ from a dense subset of Pac (H0 )H. We pull out a resolvent RH0 (−1) to take advantage
of relative boundedness and split the space into an inner part covered by the short range assumption and
an outer part as for Perry’s estimate:
||V e±itH0 φ|| =||V RH0 (−1) exp(∓itH0 )(H0 + 1)ψ||
=||V RH0 (−1)(1 − χ2a|t| + χ2a|t| ) exp(∓itH0 )(H0 + 1)ψ||
≤||V RH0 (−1)||||(1 − χ2a|t| exp(∓itH0 )(H0 + 1)ψ||
+ ||V RH0 (−1)χ2a|t| ||||(H0 + 1)ψ||
In the second term we have exactly the short range assumption: we require that the integral exists for a
dense set of functions. The first term is a case of Perry’s estimate (note that 1 − χb = χ{x||x|<b} ), if we
restrict H0 energy and D-spectrum of our test functions by ψ = f (H0 )PD ((±R, ±∞))φ.
Such vectors ψ are dense in H and we know the the Møller operators exist:
D(Ω± ) = H.
(235)
We do not know yet whether ran(Ω± ) is P (ac) (H)D(H), but by the intertwining property we know that
H restricted to ran(Ω± ) is unitarily equivalent to H0 , and therefore the spectrum σ(Hac ) ⊇ σ(H0 ). As
V was relatively compact, the essential spectra of the two operators agree and as we have excluded the
singular spectrum, we know that σ(H)ac = σ(H0 ) = [0, ∞).
Note that we originally have formulated the Perry estimate only for free motion, but relative boundedness of the potential ensures that there is no substantial difference between interacting and free motion.
13.8.10
Compactness — in- and outgoing parts are almost free wave packets
We define P± := PD ((0, ±∞)).
Lemma 18. Suppose V is short range. Let f ∈ C0∞ ((0, ∞)), then (Ω± − 1)f (H0 )P± is compact.
This Lemma is essential for the prove. Stating that (Ω± −1)f (H0 )P± is compact shows that classifying
the components as in- or outgoing by their behavior under dilatations is meaningful: on the separate
pieces, the scattering operators differ “little” from 1 (again with the precaution of not including too small
velocities by applying f (H0 )). “Little” here means by less than except on a finite dimensional subspace.
Proof. The proof combines several arguments that we have used before. The following few are specifically
interesting (for the complete proof → Teschl).
• We will use the characterization of compactness as converting a weakly convergent series ψn * into
strongly convergent.
• We write, using the dyson equation:
Ω± − I = lim eitH e−itH0 − 1 = +i
t→±∞
Z
∞
exp(isH)V exp(−isH0 )ds
0
By this we can estimate
Z
||RH (z)(Ω± − 1)f (H0 )P± ψn || = ||
Z
≤
0
81
. . . ||
∞
RH (z)V exp(−isH0 )f (H0 )P± ψn ||ds
• To exploit compactness of RH V RH0 due to short range, we insert, as before 1 = RH0 (1)(H0 + 1)
• Existence of the integral uses the same splitting into “local” and “remote” parts as in the application of Cook’s criterion and dominated convergence. Convergence to zero is a consequence of
compactness of RH (z) − RH0 (z).
13.8.11
Final steps of the proof
What is left to show is that any scattering wave packet is ψ ∈ ranΩ+ and ψ ∈ ranΩ− . We will do this
by showing that any scattering wave packet ψ⊥ 3 ranΩ− has ||ψ⊥ || = 0
Proof.
• Let ψ ∈ Hc (H) = Hac (H) with ψ = f (H)ψ for f ∈ C0∞ ((0, ∞). I.e. ensure there are not
any too low (and neither too high) energies in the wave packet.
• Denote ψ(t) = exp(−itH)ψ. By RAGE ψ(t) → 0 for t → ±∞
• Separate the in- from the out-parts of ψ(t)
φ± (t) = f (H0 )P± ψ(t)
with P± := PD ((0, ±∞)), P+ + P− = 1 (the missing spectral point λ = 0 is zero weight as σ(D) is
ac). Note that here we use f (H0 ) for the spectral cut rather than f (H).
• In spite of the different spectral cuts, the wave packet ψ(t) with finite velocities ultimately decomposes into in- and out-packets as t → +∞:
e−itH ψ =: ψ(t) ∼ ϕ− (t) + ϕ+ (t),
ϕ± (t) ∈ P± H
(236)
Here short range and ensuing compactness of differences between functions of f (H) and f (H0 ) is
essential:
ψ(t) = φ+ (t) + φ− (t) + (f (H) − f (H0 ))ψ(t) :
as ψ(t) * 0 by compactness (f (H) − f (H0 ))ψ(t) → 0.
• Again by compactness of (1 − Ω± )f (H0 ), we have
lim ||(Ω± − 1)φ± (t)|| = 0.
t→±∞
i.e. at sufficiently large times are φ± (t) are almost invariant under the scattering operator.
• Now assume we had some ψ from the orthogonal complement ran(Ω± )⊥ . Its norm is
||ψ||2 = lim hψ(t)|ψ(t)i
t→∞
= lim hψ(t)|φ+ (t) + φ− (t)i
t→∞
= lim hψ(t)|Ω+ φ+ (t) + Ω− φ− (t)i
t→∞
• By the intertwining property we know that ran(Ω± ) and therefore also ran(Ω± )⊥ are invariant
under the time-evolution exp(−itH). So we know that ψ(t) ∈ ran(Ω± )⊥ for all times t.
82
• By that we know that for the limits t → ±∞ the component from the respective orthogonal range
ran(Ω± )⊥ cannot contribute to the norm of ψ ∈ Hac (H) and only the terms with the opposite
signs Ω± may give non-zero contributions to the norm of ||ψ||2 :
||ψ||2 = lim hψ(t)|Ω∓ φ∓ (t)i
t→±∞
= lim hP∓ f (H0 )Ω∗∓ ψ(t)|ψ(t)i
t→±∞
= lim hP∓ f (H0 )e−itH0 Ω∗∓ ψ|ψ(t)i
t→±∞
where we have used intertwining in the last step.
• The last term goes to zero by the Corrolary of Perry’s estimate if one takes the limit R → 0.
(In physics terms that profits from the fact that at sufficiently large time indeed there remain no
ingoing contributions in an ac wave packet.)
13.9
Coulomb scattering
[Reed and Simon, section XI.9]
The wave-operators for H = −∆ − 1/r and H0 = −∆ do not exist
w − lim eit(−∆−1/r) e−it(−∆) = 0
t±∞
(237)
Why? Classical scattering: the hyperbolas of the classical 1/r potential do approach straight lines
sufficiently well. However, any particle on a Coulomb hyperbola will outrun any free particle with the
same asymptotic energy, no matter how far away from the force-center one starts, specifically with the
time-dependence
r(t) = ct + d log t + O(1).
(238)
While the linear time-dependence is exactly what we expect for free motion, the logarithmic dependence
causes trouble: assume that at one point T0 in time free and Coulomb positions are close. Then a there
is always a later time T1 when the two positions differ by an arbitrary distance L. This, in the end, is
also the reason for the weak convergence → 0 of the quantum scattering operators.
How to repair? Choose a different comparison time-evolution! Do not compare to a free particle, but
to something that gets accelerated just enough to keep up with the Coulomb particle. Define Dollard
wave operators with the time-dependent Hamiltonian
HD (t) = −∆ −
1
√
θ(−4|t|∆ − 1)
2|t| −∆
and it turns out
(D)
Ω±
= s − lim eit(−∆−1/r) e−i
t→±
Rt
0
HD (s)ds
(239)
(240)
exists. Again Volker Enss (1979) proved asymptotic completeness for Dollard comparison operators (not
in the 1972 Reed and Simon book).
Apart from the technical complications, does HD define useful dynamics for the comparison? It turns
out yes, as momentum information that we are interested in is unaltered. The long time limit of momenta
is the same as for free motion, this is what is usually measured. What changes is the exact interpretation
of arrival times at the detector, but that is usually not considered. Actually, it is hard to observe as in
a typical scattering experiment there is no information on when exactly the interaction takes place, and
therefore how long time the particle traveled before it hit the detector. (Note the arrival times can be
measured very accurately.)
83
13.9.1
Exact solutions
Like in the classical case, we also know the exact scattering solutions of the quantum Coulomb problem.
And we know the mapping of ingoing to outgoing asymptotic momenta. In that sense the single particle
Coulomb scattering problem can be considered as solved.
The reason for this solvability is the same as in the classical case: high symmetry of the system.
Except for angular momentum and energy, there is one more conserved vector: the Lenz-Runge vector:
the “perihelium” position in classical mechanics.
It turns out that “asymptotic momentum operators” p± can be defined (compare the classical hyperbolas), in terms of which HPac = p2± . After that, the scattering problem can be solved with purely
algebraic methods (for details, see Thirring).
13.9.2
Coulomb + finite range potentials
As the Coulomb dynamics is known completely, we can use it instead of the free motion as a comparison
time evolution for dynamics, that are long range Coulomb but have short range modifications. Think of
the scattering from the ion of a molecule.
Suppose we have a Hamiltonian of the form
H = −∆ − 1/r + V = Hc + V
(241)
where V (Hc − i)−1 is bounded and short range. Then the same apparatus of scattering theory can be
run for the wave operators
(c)
(242)
Ω± = lim eitH e−itHc
t→±∞
which will map H(Hc )ac → H(H)ac . Existence of the scattering operators is proven (e.g. Muleherin and
(c)
Zinnes, 1970). Asymptotic completeness ranΩ± = H(H)ac , can be inferred from the result of Enss the
general asymptotic completeness for long-range (∼ 1/r) potentials relative to Dollard operators.
14
Formal derivations
In this section we do not even try to give rigorous results. Rather, we show a few manipulations that
relate the rigorous results of the previous section to what physicists often do.
14.1
The Lippmann-Schwinger equation in disguise
The LS equation is of great practical importance in physics. We introduced it as an equation for the
2
(±)
(±)
“scattering eigenfunctions” HΨ~ = k2 Ψ~ . Unfortunately, rigorous foundation of “scattering eigenk
k
functions” and derivation of related formulae of practical importance requires the buildup of a significant
mathematical apparatus. Here we will introduce the “dirty” physicist’s formal manipulations that lead
from the scattering operators to the LS equation and further on to cross-sections.
~
Take an arbitrarily narrow wave free wave packet “ϕ~k ∼ eik·~r ”, then
|Ψ(±) i := Ω± |ϕ~k i
(243)
is indeed a seemingly much simpler equation than the LS equation to obtain scattering states Ψ(±) . The
Ψ(+) is what we want as an “in” state: in the far past it looks like the free wave packet ψ~k . The somewhat
awkward sign convention where the far past corresponds to + has historic reasons. It was not originally
chosen for the time limits. Rather, in the LS equation, the sign k 2 /2 + i corresponds to complex energies
in the imaginary upper half-plane, which is the part relevant for the time-limit to the past.
The similarity becomes even more evocative, when we will show that in the spectral representation
of H0 we have
Z
∞
(H − E ∓ ı)−1 V δ(H0 − E)dE
Ω± = 1 − lim
e↓
0
84
(244)
We use the notation for the spectral measure dP (E) = δ(H0 − E)dE (we have a pure ac spectrum and
therefor no trouble with possible point spectrum. Some signs need adjustment, but most importantly we
have the resolvent of H, not H0 . Unfortunately, this makes this formula useless for computations. But
it can be transformed to the LS equation.
14.2
Ω± in the spectral representation of H0
Ω± are time-independent operators operators obtained as time limits of time-dependent operators. Here
we will replace the limit in time by a limit of resolvents of H0 . This is achieved by using a result by Abel:
Let f be a bounded function on (0, ∞) and suppose limt→∞ f (t) = f∞ . Then
Z ∞
lim e−t f (t)dt = lim f (t).
(245)
↓0
t→∞
0
Problem 14.56: Abel limit Show that
Z
V (t) → V∞
⇒
lim ↓0
∞
e−t V (t)dt = V∞
0
as a strong limit. In the same way we can evaluate the limits of eitH e−itH0 , where we heavily rely on the
intuition that we can treat operators just like numbers as long as we respect their non-commuting nature
(and take care of domain questions).
In the spectral representation of H0
Z ∞
H0 =
EdP (E)
(246)
0
we write
e
itH −itH0
e
=e
itH
Z
∞
e
−itE
Z
dP (E) =
0
∞
eit(H−E) dP (E)
(247)
0
The integrals exists, as the involved functions and operators are bounded. Now we use the Abel method
and formally the time-integral
Z ∞
Z ∞
Ω± = lim e−t
e±it(H−E) dP (E)dt
(248)
↓0
0
0
Z ∞Z ∞
= lim e±it(H−E±i) dt dP (E)
(249)
↓0
0
0
Z ∞
= lim ±i
(H − E ± i)−1 dP (E)
(250)
↓0
0
Z ∞
= 1 − lim
(H − E ± i)−1 V dP (E)
(251)
↓0
0
The last step is by some algebra with the resolvent
(H − E ± i)−1 (±i)
=
(H − E ± i)−1 [(H − E ± i) − (H − E)]
=
1 − (H − E ± i)−1 (H0 + V − E)
(252)
and observing that
(H0 − E)dP (E) = (H0 − E)δ(H0 − E)dE = 0.
For the time integral we have treated H like an ordinary function: as it is selfadjoint, it has a spectral
representation
Z
H = GdPH (G)
(253)
85
for the projection valued measure PH belonging to H. Then
Z ∞
Z Z ∞
e±it(H−E±i) dt =
e±it(G−E±i) dP (G)dt
0
(254)
0
with G ∈ σ(H) ⊂ R. Integrals can be interchanged, the time integral can be performed and the spectral
representation can be replaced by the abstract operator again.
Thus we arrive at
Z ∞
Ψ(±) = Ω± φ = φ − lim
(H − E ± i)−1 V dP (E)φ,
(255)
↓0
0
which bears great similarity with the LS equation. Actually, it is seemingly better, as it is a direct
expression for some scattering wave packet Ψ(±) rather than an integral equation for it. However, it
contains (H − E ± i)−1 , which usually is rather inaccessible to computation.
Now we remember the symmetric role of H and H0 in scattering theory and we interchange them
H, H0 , H − H0 =: V → H 0 = H0 , H00 = H, H 0 − H00 = H0 − H = −V
to write
φ
(±)
Z
= ψ + lim
↓0
(256)
∞
(H0 − E ± i)−1 V dPH (E)ψ,
ψ ∈ Pac (H)H.
(257)
0
Next remember that the signs ± just indicate whether the φ(±) and ψ are mapped by going through the
the remote past or future. That is, we can just as well move ± from φ back to ψ, leading to
Z ∞
(±)
ψ
= φ − lim
(H0 − E ± i)−1 V dPH (E)ψ (±) .
(258)
↓0
14.2.1
0
Remark on “ac-spectral eigenfunctions”
The equation above is almost the shape of the LS equation. However, it is formulated for wave-packets
rather than, as physicists like to do it, for “eigenfunctions of the continuous spectrum”. For −∆ these
would be plane waves exp(i~k · ~x). Indeed, the path for making this notion precise, is through the Fourier
transform and using the mapping by Ω± , formally
~
(±)
|~kiH = Ω± |~kiH0 := Ω± eik·~x .
(259)
The mapping is not unique because of the two alternative signs, but it does not need to be, as the energies
are expected to be degenerate. Time-reversal maps between the two signs. Accepting that, we can even
give meaning to the δ-function of the Hamiltonian as the spectral projector onto the eigenfunctions of
given energy
Z ∞
Z
k
(260)
δ(H − E)dE :=
dαdk 2 |k, αihk, α|
2
0
Z ∞
Z
√
E 1/2 √
=
dE dα
| E, αih E, α|
(261)
2
Z0 ∞
Z
^
^
=
dE dα|E,
αihE,
α|
(262)
0
where
E 1/4 √
^
(263)
|E,
αi = √ | E, αi.
2
If we do that, formally, the δ(H −E) allows to perform the dE integration and we have the LippmannSchwinger equation for the scattering solutions
ψ (±) (E) = φ(E) − lim(H0 − E ± i)−1 V ψ (±) (E).
↓0
~
(264)
where the + corresponds to matching of asymptotics of φ = eik~r in the remote future (“outgoing boundary
conditions”) and − to matching in the remote past (“ingoing boundary conditions”).
86
14.2.2
The LS equation in position space
In position space, the LS equation takes the form
ψ
(±)
i~
k~
r
(~r) = e
0
Z
−
dr
(3)
R3
e∓ik|~r−~r |
V (~r)ψ (±) (~r0 ).
4π|r − r0 |
(265)
where we recognize the form of the solution ψ (±) (~r) as a plane wave plus a superposition of spherical
waves.
Problem 14.57: Green’s function in position space Compute the function
G± (~r, ~r0 ) =
1
(2π)3
Z
~
d(3) k
0
0
eik(~r−~r )
1 e±ik0 |~r−~r |
=−
2
2
k0 − k ± i
4π |~r − ~r0 |
(We use the physicists’ convention not to explicitly write the lim ).
Hint: Use polar coordinates, expansion of a plane wave into spherical Bessel functions, integrate over
the angle, use the residue theorem for what remains.
14.3
Mapping free to free spaces: the S-matrix
With asymptotic completeness the Ω± unitary bijections Pac (H)H ↔ Pac (H0 )H and the S-matrix maps
Pac (H0 )H → Pac (H0 )H
S = (Ω+ )−1 Ω− = Ω∗+ Ω−
(266)
This corresponds to a limit
lim [U (−t)U0 (t)]−1 [U (t)U0 (−t)] = lim U0 (−t)U (2t)U0 (−t)]
t→∞
t→∞
(267)
i.e. bring a free packet sufficiently far into the past, let it interact for sufficiently long time (into the
future), propagate the free packet it back into the present: generate the effect of scattering as a mapping
between free wave packets.
Some properties of the S-matrix:
• is unitary (asymptotic completeness)
• it “inherits” symmetries from H and H0 .
• it conserves energy - intertwining relation
14.3.1
Spectral representation of the S-matrix
In the same spirit as for Ω± , there is a representation of the S-matrix in the spectral representation of
H0 :
Z ∞
S = lim
dE 1 − 2πıδ(H0 − E) V − V (H − E − ı)−1 V δ(H0 − E).
(268)
↓0
0
We use the (only slightly) symbolic notation
δ(H0 − E) =
Z
X
|E, αihE, α|,
α
where α labels the degeneracy in the energy subspace E.
Energy conservation follows from the intertwining property of the Møller operators
−1
−1
SH0 = Ω−1
+ Ω− H0 = Ω+ HΩ− = H0 Ω+ Ω− = H0 S.
87
(269)
It implies that S is a function of H0 , i.e. in the spectral representation is a multiplication by a function
of E. It may still connect different α, though (after all, we are free to choose
L the basis in the space of
fixed E spanned by the |E, αi. In the rotationally symmetric case, S =
lm Sl we know that l, m is
not changed
and
therefore
can
be
parameterized
by
the
“scattering
phases”
σ
(k), conventionally written
l
√
with k = E:
S|E, l, mi = e2iδl (k) |E, l, mi
(270)
Note that for rotationally symmetric problems there cannot be any dependence on m.
The label α can be discrete or continuous. Think of polar angles to denote directions, or, alternatively,
the corresponding resolution of the rotations in terms of sphereical harmonics:
X
∗
|αihα| = |φ, θihφ, θ| = P (φ, θ) =
Ylm (φ, θ)Ylm
(φ0 , θ0 ) = δ(θ − θ0 )δ(φ − φ0 )
(271)
lm
14.3.2
The T -matrix
Let ψ, ψ be two wave packets. The matrix elements of the S-matrix can be written as
Z
hψ, Sψi = hψ, ψi − 2iπ d(3) k ψ(~k)∗ T (~k, ~k 0 )δ(~k 2 − ~k 02 )ψ(~k 0 )
(272)
where ψ(~k) is the representation of ψ in ~k-space with the T-matrix
T (~k, ~k 0 ) = h~k|V − V (H − E − ı)−1 V |~k 0 i,
(273)
where |~ki = exp(i~k~x)/(2π)3/2 are the δ-normialized plane waves h~k|~k 0 i = δ (3) (~k − ~k 0 ) and ψ(k) = h~k|ψi.
T (~k, ~k 0 ) contains the relevant scattering information. Also, it is a smooth function of ~k, ~k 0 , loosely
speaking, where solutions of the LS equation exist. It essentially is the same as the scattering amplitude
f (k, ~n, ~n0 )
f (k, ~n, ~n0 ) = −(2π)2 T (k~n, k~n0 )
(274)
whose square gives the differential cross section:
σ(~k, ~k 0 ) = |f (k, ~n, ~n0 )|2 .
14.4
(275)
Scattering cross section — derivation
Notation In the following, for easier connecting to standard literature, we assume mass =1, i.e. H0 =
−∆/2 and the E = k 2 /2.
14.4.1
Post and prior form
Note that on the level of generalized eigenfunctions/the LS equation we have
(±) ~
2
−1
|ψ (k)i = Ω± φ~k = 1 − lim(H − k ± i) V |φ~k i
↓0
(276)
and we recover the known formulae of scattering theory
−
1
f (~k, ~k 0 )
2π 2
= hφ~k |V − V (H − k 2 + i)−1 V |φ~k0 i
(+)
k
(278)
|φ~k0 i
(279)
= hφ~k |V Ω+ |φ~k0 i = hφ~k |V |ψ~ 0 i
= hΩ− φ~k |V φ~k0 i =
(−)
hψ~ |V
k
(277)
These are the post(+) and prior(-) form, respectively, of the scattering amplitude. The two form are
rigorously equal. In practice the distinction can become important in a multi-channel situation, where
88
the left two scattering solutions ψ (+) and ψ (−) differ and approximations to one of them can be more
appropriate than to the other.
Problem 14.58: First Born approximation Compute the scattering amplitude for the
Yukawa potential
V (~x) = V0
e−µ|~x|
,
µ|~x|
µ>0
(280)
in “first Born approximation”, expressing ψ (±) by its the first term of the Born series. Express results as
a function of cos γ, where cos γ = ~k~x/(|~k||~x|), γ the angle between ~k and ~x.
(a) How does the first Born cross section depend on the sign of V0 ?
(b) How does the cross section behave with |~k| → 0?
(c) What is the total cross section
Z
1
Z
dφ|f (1) (~k, ~k0 )|2 ?
d cos θ
(281)
−1
θ and φ are the polar angles of ~k = |~k0 |(sin θ cos φ, sin θ sin φ, cos θ)
Hint: Use the expansion of a plane wave into its spherical components
X
~
eık~x =
(2l + 1)Pl (cos γ)ıl jl (|~k||~x|)
(282)
l
where the Pl are Legendre polynomials.
P0 = 1,
P1 (x) = x,
P2 (x) =
1
(3x2 − 1) . . .
2
(283)
with the spherical Bessel functions
j0 (x)
=
j1 (x)
=
j2 (x)
=
sin(x)
x
sin(x) cos(x)
−
2
x
x
3
sin(x) 3 cos(x)
−1
−
x2
x
x2
...
(284)
(285)
(286)
(287)
89