Random Operators
Lecture Notes by
Michael Aizenman & Simone Warzel
Preliminary draft
(with many misprints)
D EPARTMENT OF M ATHEMATICS AND P HYSICS ,
VERSITY, P RINCETON , NJ 08544, USA
P RINCETON U NI -
Z ENTRUM M ATHEMATIK , TU M ÜNCHEN , B OLTZMANNSTR . 3,
85747 G ARCHING , G ERMANY
Contents
Chapter 2. Some mathematical groundwork
1. Elements of spectral theory
1.1. Hilbert spaces, linear operators and all that
1.2. Spectral calculus and spectral types
2. Spectra and dynamics
2.1. Return probabilities and recurrence
2.2. Theorems of Wiener and RAGE
2.3. Theorems of Strichartz and Last
3. Appendix: Herglotz functions
Notes
5
5
5
8
11
11
12
15
16
17
Chapter 3. Modeling disorder
1. Stochastic processes
1.1. Basic notions
1.2. Ergodicity
2. Ergodic operators
2.1. Definitions
2.2. Deterministic spectra
2.3. Determining the spectrum
Notes
19
19
19
20
22
22
23
24
26
Bibliography
29
3
CHAPTER 2
Some mathematical groundwork
1. Elements of spectral theory
Since observables in quantum mechanics are described by self-adjoint
operators on a Hilbert space, we will briefly and in a somewhat informal
manner review some elements of the theory of such operators. We will omit
all proofs and refer the reader to the extended literature, in particular [146,
41, 128, 153].
1.1. Hilbert spaces, linear operators and all that. The arena of quantum mechanics is a separable complex Hilbert space H, which is endowed
with a scalar product h·, ·i : H × H → C. We will take the latter to
be linearpin its second entry. The induced norm of ψ ∈ H is given by
kψk := hψ, ψi.
An example is the space of square-summable, complex valued sequences over the d-dimensional square lattice
n
o
X
`2 (Zd ) := ψ : Zd → Zd |ψ(x)|2 < ∞ ,
(2.1)
x∈Zd
which is rendered a Hilbert space together with the scalar product
X
ϕ(x) ψ(x) .
hϕ, ψi :=
(2.2)
x∈Zd
Here (·) denotes complex conjugation.
A linear operator A : D(A) → H defined on a domain, i.e., a dense
linear subspace D(A) ⊂ H, is bounded if and only if there exists some
a < ∞ such that kAψk ≤ akψk for all ψ ∈ D(A). The smallest possible
constant in this inequality defines the operator norm
kAk := sup kAψk .
ψ∈D(A)
kψk=1
5
6
2. SOME MATHEMATICAL GROUNDWORK
An example of a bounded operator is the discrete Laplacian
∆ : `2 (Zd ) → `2 (Zd ) ,
X
ψ(x) :=
(ψ(y) − ψ(x))
(2.3)
y∈Zd
|x−y|=1
defined on `2 (Zd ). Its operator norm is estimated using the triangle inequality for the norm on `2 (Zd ):
2 1/2
XX
k∆ψk =
(ψ(x + j) − ψ(x)) x∈Zd
≤
|j|=1
X h X
|j|=1
2
|ψ(x + j)|
1/2
+
x∈Zd
X
2
|ψ(x)|
1/2 i
x∈Zd
= 4d kψk .
(2.4)
As a consequence, k∆k ≤ 4d. As we will see below, this inequality is, in
fact, an equality.
The space of linear operators on H may be equipped with various
topologies. For sequences of bounded operators the most important concepts are those of weak, strong and norm convergence, i.e.,
An → A weakly
iff hϕ , (An − A) ψi → 0 for all ϕ, ψ ∈ H,
An → A strongly
iff
k(An − A) ψk → 0 for all ψ ∈ H,
An → A in norm
iff
kAn − Ak → 0 .
Clearly, norm convergence implies strong convergence which in turn implies weak convergence.
We will mainly be concerned with self-adjoint operators. Recall that
the adjoint of A : D(A) → H is the unique operator A† : D(A† ) → H
acting as
hϕ, Aψi = hA† ϕ, ψi
for all ψ ∈ D(A) and ϕ ∈ D(A† ) := {ϕ ∈ H | There is % ∈ H such that for
all ψ ∈ D(A): hϕ, Aψi = h%, ψi}. The operator A is self-adjoint if and
only if A = A† , which in particular requires D(A) = D(A† ).
Any bounded, real-valued sequence, V : Zd → R, gives rise to a
bounded and self-adjoint Schrödinger operator H : `2 (Zd ) → `2 (Zd )
defined by
(Hψ) (x) := − (∆ψ) (x) + V (x) ψ(x)
(2.5)
1. ELEMENTS OF SPECTRAL THEORY
7
on ψ ∈ `2 (Zd ). In fact, each of the terms individually define bounded
self-adjoint operators and hence their sum is bounded and self-adjoint. The
self-adjointness of the multiplication operator V : `2 (Zd ) → `2 (Zd ) corresponding to V is left as an easy exercise to the reader. The self-adjointness
of the discrete Laplacian follows from its representation as a quadratic
form,
1 X
hϕ, −∆ψi =
(ϕ(y) − ϕ(x)) (ψ(y) − ψ(x)) ,
(2.6)
2
d
x,y∈Z
|x−y|=1
which implies hϕ, −∆ψi = h−∆ϕ, ψi for all ϕ, ψ ∈ `2 (Zd ).
We recall that the resolvent set of an operator A : D(A) → H is
given by
%(A) := {z ∈ C | (A − z) : D(A) → H is bijective} .
For z ∈ %(A) the operator (A − z) is hence invertible and its inverse, (A −
z)−1 : H → D(A) is called the resolvent. The spectrum of A is the set
σ(A) := C \ %(A) ,
which is always closed. If A is bounded, σ(A) is compact. In case A is
self-adjoint, σ(A) ⊂ R and for all z ∈ %(A):
1
1
(A − z)−1 ≤
≤
,
(2.7)
dist(z, σ(A))
| Im z|
where the last inequality requires z ∈ C\R.
As an example, let us determine the spectrum of the discrete Laplacian. The Fourier transformation
F : `2 (Zd ) → L2 ([0, 2π]d )
X
1
(Fψ) (k) :=
e−ik·x ψ(x)
d/2
(2π)
d
(2.8)
x∈Z
defines a unitary operator. By an explicit calculation, one checks that the
discrete Laplacian is unitarily equaivalent to a multiplication operator, i.e.
F∆F
−1
d
X
ϕ (k) = 2
(cos(kν ) − 1) ϕ(k) =: h(k) ϕ(k) ,
(2.9)
ν=1
for all φ ∈ L2 ([0, 2π]d ), where k = (k1 , . . . , kd ) ∈ [0, 2π]d . The spectrum
of any multiplication operator is the range of the corresponding function,
8
2. SOME MATHEMATICAL GROUNDWORK
which in case of h, is the interval [−4d, 0]. Since the spectrum is invariant
under unitary transformation, we therefore conclude
σ(−∆) = [0, 4d] .
(2.10)
This implies k∆k = 4d, since the operator norm coincides with the maximal modulus of points in the spectrum.
An important tool for spectral analysis are the first and second resolvent equation.
Proposition 2.1 (Resolvent equations) For a linear operator A : D(A) →
H and z1 , z2 ∈ %(A):
(A − z1 )−1 − (A − z2 )−1 = (z1 − z2 )(A − z1 )−1 (A − z2 )−1 .
(2.11)
Moreover, if B : D(B) → H is defined on a common domain, D(A) =
D(B),
(A − z)−1 − (B − z)−1 = (A − z)−1 (B − A) (B − z)−1
= (B − z)−1 (B − A) (A − z)−1 ,
(2.12)
for any z ∈ %(A) ∩ %(B).
Lecture 1
21.4.10
The proof is elementary and can be found in [153].
An immediate corrollary is the fact that the distance of the spectra of
two operators A, B : D(A) → H whose difference is a bounded operator is
bounded by
dist (σ(A), σ(B)) ≤ kA − Bk .
(2.13)
1.2. Spectral calculus and spectral types. The benefit of dealing with
self-adjoint operators A : D(A) → H on a Hilbert space is the availablity
of spectral calculus.
Recall that for any ψ ∈ H, the function given by
Fψ (z) := hψ , (A − z)−1 ψi
(2.14)
is holomorphic for z ∈ %(A) ⊃ C+ := {z ∈ C | Im z > 0}. Using (2.7)
we conclude:
kψk2
|Fψ (z)| ≤ (A − z)−1 kψk2 ≤
for any z ∈ C+
Im z
Fψ (z) = h(A − z)−1 ψ , ψi = hψ , (A − z)−1 ψi = Fψ (z) .
Moreover, the first resolvent equation (2.11) implies:
2
1 Im Fψ (z) =
Fψ (z) − Fψ (z) = Im z (A − z)−1 ψ > 0 ,
2i
1. ELEMENTS OF SPECTRAL THEORY
9
for any ψ 6= 0, hence Fψ is a Herglotz function, i.e., a holomorphic function Fψ : C+ → C+ mapping the upper half plane into itself.
The representation theorem for Herglotz functions, which is summarized in the appendix, entails that there exists a unique finite Borel measure
µψ on R, called the spectral measure of A associated with ψ, such that
Z
µψ (dλ)
Fψ (z) =
, z ∈ C+ .
(2.15)
λ−z
By polarization, for any pair ϕ, ψ ∈ H one may identify
complex finite
R
Borel measures µϕ,ψ such that hϕ, (A − z)−1 ψi = (λ − z)−1 µϕ,ψ (dλ).
These measures enable the definition of functions f (A) of a self-adjoint
operators, i.e., for any f ∈ L∞ (R):
Z
hϕ, f (A) ψi :=
f (λ) µϕ,ψ (dλ) .
(2.16)
This implicates the following calculus for functions f, g ∈ L∞ (R) of selfadjoint operators:
(f + g)(A) = f (A) + g(A) , (f g)(A) = f (A) g(A) , f (A) = f (A)† ,
If f ≥ 0, then
hψ, f (A)ψi ≥ 0
for all ψ ∈ H.
Moreover, kf (A)k = supλ∈σ(A) |f (λ)|.
Of particular importance are indicator functions,
1 if x ∈ J,
1J (x) :=
, J ⊂ R.
0 else.
They define orthogonal projections PJ (A) := 1J (A). In fact, the map J 7→
PJ (A) on the Borel sets J ⊂ R defines a projection-valued measure, i.e.
(1) PR (A) =S1 and P∅ (A) = 0
(2) P
If J = n Jn with Jn ∩ Jm = ∅ for n 6= m, then PJ (A) =
n PJn (A).
The support of this measure coincides with the spectrum,
σ(A) = λ ∈ R P(λ−ε,λ+ε) (A) 6= 0 for all ε > 0. .
(2.17)
The spectral theorem ensures that there is one-to-one correspondence between self-adjoint operators and projections valued measures. This is summarized by (2.16) through the relation hϕ , PJ (A) ψi = µϕ,ψ (J).
Every Borel measure µ can be uniquely decomposed with respect to
the Lebesgue measure into three mutually singular parts,
µ = µpp + µsc + µac .
10
2. SOME MATHEMATICAL GROUNDWORK
Whereas µac is absolutely continuous with respect to Lebesgue measure, the
sum of the first two terms is singular. It consists of a pure point component,
µpp , and a singular continuous remainder, µsc .
Accordingly, for a self-adjoint operator A : D(A) → H one decomposes the Hilbert space into closed subspaces,
o
n
#
#
H := ψ ∈ H | µψ = µψ , # = pp, sc, ac
This decomposition turns out to be orthogonal, H = Hpp ⊕ Hsc ⊕ Hac and
the above subspaces are left invariant under the action of A.
In case of a bounded self-adjoint operator A : H → H, the restriction
to these subspaces defines the components of the spectrum,
σ # (A) := σ A # , # = pp, sc, ac ,
(2.18)
H
which are called the pure point, singular continuous and absolutely continuous spectrum. The point spectrum coincides with the closure of the set
of eigenvalues,
σ pp (A) = {λ ∈ R | λ is an eigenvalue of A} .
Let us conclude this subsection with two examples.
For a multiplication operator corresponding to a real-valued sequence
V ∈ `∞ (Zd ), one identifies the spectral measure corresponding to ψ ∈
`2 (Zd ) as a weighted sum of Dirac measures:
X
µψ =
|ψ(x)|2 δ{V (x)} ,
x∈Zd
i.e., µψ = µpp
ψ . The eigenvectors are given by the localized vectors δx ∈
2
d
` (Z ),
1 if x ∈ J,
δx (y) :=
(2.19)
0
else.
The corresponding eigenvalues are {V (x) | x ∈ Zd }, whose closure is the
pure point spectrum σ pp (V ).
For the Laplacian on `2 (Zd ) we use the unitarily equaivalent representation as multiplication operator on L2 ([0, 2π]d ) given in (2.9) to determine
the spectral measure:
hψ, f (∆)ψi = hFψ , F f (∆) F −1 Fψi = hFψ , f F∆F −1 Fψi
Z
Z
2
=
|(Fψ) (k)| f (h(k)) dk =
f (λ) µψ (dλ) . (2.20)
[0,2π]d
2. lecture (incl.
Herglotz App.)
23.4.2010
2
d
Hence, µψ = µac
ψ for all ψ ∈ ` (Z ) and the spectrum is only absolutely
continuous, σ ac (−∆) = [0, 4d].
2. SPECTRA AND DYNAMICS
11
2. Spectra and dynamics
The quantum time evolution generated by a self-adjoint operator H
on some Hilbert space H is given by a unitary group of operators,
ψ(t) := e−itH ψ ,
t ∈ R.
(2.21)
As we shall now see, the dynamical properties of the evolution are closely
related with the spectral characteristics of H.
2.1. Return probabilities and recurrence. For a relation between the
two, consider the probability |hψ, ψ(t)i|2 of a system in a state ψ ∈ H
at time t = 0 to return to itself at time t > 0. This probability may be
expressed in terms of the Fourier transform
Z
µ̂ψ (t) :=
e−itE µψ (dE) = hψ, ψ(t)i
(2.22)
of the spectral measure of H associated with ψ.
More generally, quantum observables are given by self adjoint operators A, with the range of possible outcomes of the measurement given by
operator’s spectrum. For a quantum system which at time t = 0 is in the
state ψ ∈ H and for which an observable A is measured at time t > 0 the
probability of observing a value in the range I ⊂ R, is given by:
Prob (A(t) ∈ I | ψ) := kPI (A) ψ(t)k2
where PI (A) is the spectral projection operator. In the basic example with A
a rank-one projection operator onto φ, the above reduces to the expression:
Z
2
hφ, e−itH ψi2 = e−itE µφ,ψ (dE) = |µ̂ψ,φ (t)|2 ,
(2.23)
which is again given by a Fourier transform, namely, of the spectral measure
µψ,φ associated with φ, ψ ∈ H, which in general is complex.
It is a simple, but perhaps somewhat astounding observation that for
finite quantum systems, with dim H < ∞, the above expressions yield
quasi-periodic function of time. Denoting by (En ) the eigenvalues of H,
and by (ξn ) the corresponding eigenvectors:
X
Prob (A(t) ∈ I | ψ) =
eit(En −Em ) hψ , ξn i hξn , PI (A) ξm ihξm , ψi .
n,m
The fact that as t → ∞ this function does not converge to an asymptotic
value, and repeatedly assumes values arbitrarily close to kPI (A) ψk2 is reminiscent of the Poincaré recurrence phenomenon of classical mechanics of
finite systems. There, the time evolution is described by measure preserving flows in a classical phase space. If taken literarily, the recurrence has
12
2. SOME MATHEMATICAL GROUNDWORK
the implication that if gas is released into a room from a bottle, then, with
probability one, there will be a moment when all the gas will be found back
in the bottle. Of course, the recurrence times for a macroscopic system,
for which the number of molecules can be estimated through the Avogadro
number NA ≈ 6 · 1023 mol−1 , is so long that well before that the door
will be opened, rendering the model insufficient. Actually, by the typical
recurrence time for such an event far more greviuous deviations from the
idealized description will occur (even the lab will no longer be there).
2.2. Theorems of Wiener and RAGE. Recurrence is avoidable in infinite models, e.g., with H = `2 (Zd ), for which a key observation is that relaxation in time does occur in the presence of continuous spectrum. In case
of absolutely continuous (ac) spectrum, the Riemann-Lebesgue lemma
implies the pointwise decay of the Fourier transform of the ac component
of the spectral measure:
lim µ̂ac
(2.24)
ψ (t) = 0 .
t→∞
Relaxation in case of more general continuous measures follows from a
celebrated theorem of N. Wiener.
Theorem 2.2 (Wiener) Let µ be a finite complex Borel measure on R.
Then
Z
X
1 T
lim
|µ̂(t)|2 dt =
|µ({E})|2 .
(2.25)
T →∞ T 0
pp
E∈supp µ
Proof: The assertion follows by Fubini’s theorem:
Z Z Z T
Z
1 T
1
2
i(E−E 0 )t
|µ̂(t)| dt =
e
dt µ(dE 0 ) µ(dE)
T 0
T 0
Z Z iT (E−E 0 )
e
−1
=
µ(dE 0 ) µ(dE) .
(2.26)
0
i(E − E )T
Since the integrand is uniformly bounded by one and converges pointwise
to 1{0} (E − E 0 ), the dominated convergence theorem implies
Z
Z Z
1 T
2
lim
|µ̂(t)| dt =
1{0} (E − E 0 ) µ(dE 0 ) µ(dE)
T →∞ T 0
Z
X
=
µ({E}) µ(dE) =
|µ({E})|2 . (2.27)
E∈supp µpp
2. SPECTRA AND DYNAMICS
13
Corollary 2.3 Let H be a self-adjoint and A be a compact operator on
some Hilbert space H. Then
1
lim
T →∞ T
Z
T
−itH 2
A e
ψ dt = 0
(2.28)
0
for all initial states ψ ∈ Hc := Hsc ⊕ Hac , for which the spectral measure
of H is continuous.
Proof: In case A is a rank-one operator, i.e., A = |ρihφ|, the claim follows
from Wiener’s theorem (Theorem 2.2) applied to the spectral measure µφ,ψ ,
cf. (2.23). As a consequence it also holds for finite-rank operators A. Any
compact operator A may in turn be approximated by finite-rank operators.
More precisely, for every ε > 0 there exists A of finite rank such that
kA − Aε k < ε and hence
kA ψ(t)k ≤ kAε ψ(t)k + k(A − Aε ) ψ(t)k ≤ kAε ψ(t)k + ε kψk . (2.29)
The Césaro average of the first term goes to zero in the long-time limit. This
completes the proof since ε was arbitrary.
The Wiener theorem implies the following dynamical characterization of the subspaces of H associated with the continuous and pure point
part of the generator H of the time evolution. The result is named after
D. Ruelle [131], W. Amrein, V. Gorgescu [10] and V. Enss [46].
Theorem 2.4 (RAGE) Let H be a self-adjoint operator on some Hilbert
space H and An be a sequence of compact operators which converge strongly
to the identity. Then
H = ψ∈H
pp
H = ψ∈H
c
Z T
2
1
−itH
lim lim
An e
ψ dt = 0
n→∞ T →∞ T 0
lim sup (1 − An )e−itH ψ = 0
n→∞ t∈R
Proof: If ψ ∈ Hc the previous corollary implies limT →∞
0 for every n. If ψ ∈ Hpp we claim
lim sup k(1 − An )ψ(t)k = 0 .
n→∞ t∈R
1
T
RT
0
(2.30)
(2.31)
kAn ψ(t)k2 dt =
(2.32)
14
2. SOME MATHEMATICAL GROUNDWORK
For a proof we expand ψ into eigenfunctions (ξn ) of H, and split the sum
into the first N terms and a remainder with norm less than ε,
ψ=
N
X
hξk , ψi ξk + φN ,
kφN k < ε .
(2.33)
k=1
Since limn→∞ k(1 − An )ξk k = 0 for every k, the first term contributes zero
to limit in (2.32). This completes the proof of (2.32) since ε may be chosen
arbitrarily small.
Since every ψ may be uniquely decomposed into a component ψ c ∈
Hc and an orthogonal one ψ pp ∈ Hpp , it remains to show that (i) the limit
in (2.30) does not tend to zero for ψ pp and, (ii) the limit in (2.31) does not
tend to zero for ψc .
The proof of the first assertion relies on (2.32) which implies that
kAn ψ pp (t)k ≥ |kψk − k(1 − An )ψ pp (t)k| > 0
(2.34)
for all t ≥ 0 and sufficiently large n.
The second assertion follows by contradiction. Suppose (2.32) applies to ψ c , then Corollary 2.3 implies
21
Z
1 T
2
c
k(1 − An )ψ (t)k dt
0 = lim lim
n→0 T →∞ T 0
21
Z T
1
2
≥ kψ c k − lim lim
kAn ψ c (t)k dt
= kψ c k
(2.35)
n→0 T →∞ T 0
which contradicts ψ c 6= 0.
In case H = `2 (Zd ) one may pick (An ) as indicator functions of balls.
Eqs. (2.30) and (2.31) then express the fact that extended states ψ c ∈ Hc
are characterized by eventually leaving every ball,
Z
1 T X
lim lim
|hδx , ψ c (t)i|2 dt = 0 ,
(2.36)
R→∞ T →∞ T 0
|x|≤R
pp
and bound states ψ pp ∈ H by being forever confined to a sufficiently large
ball
X
lim sup
|hδx , ψ pp (t)i|2 = 0 .
(2.37)
R→∞ t∈R
|x|≤R
Similar interpretations exist for quantum systems on a continuous manifold,
e.g., H = L2 (Rd ). In this context, it is worth noting that Corollary 2.3 as
well as the RAGE theorem can be extended to operators A which are relatively compact with respect to H. We refer the interested reader to [146].
2. SPECTRA AND DYNAMICS
15
2.3. Theorems of Strichartz and Last. Wiener’s theorem implies that
upon Césaro average the return probability converges to zero for states with
purely continuous spectral measure. More can be said about the rate of
convergence for the following class of continuous measures.
Definition 2.5 Let µ be a finite Borel measure on R and α ∈ [0, 1]. Then µ
is called uniformly α Hölder continuous (UαH) if there is some C < ∞
such that for all intervals I with |I| < 1 one has
µ(I) ≤ C |I|α .
(2.38)
In case α ∈ (0, 1) the above measures are singular continuous. In
physical applications such measures occur in models with quasi-period potentials. Prominent examples are the Harper Hamiltonian at criticality or the
Fibbonacci Hamiltonian. For these examples, the decay rate in the following theorem has been argued for by the authors of [64], who were apparently
unaware of Strichartz’ work [143].
Theorem 2.6 (Strichartz) Let µ be UαH for some α. Then there exists
some C < ∞ such that for all f ∈ L2 (R, µ)
Z
Z
1 T c 2
C
|f (E)|2 µ(dE) .
(2.39)
f µ(t) dt ≤ α
T 0
T
Proof: We estimate the indicator function of [0, T ] by a Gaussian, and
compute the Fourier transform
Z
Z
2
2 e
1 T c 2
− t2 c
T
e
f µ(t) ≤
f µ(t) dt
T 0
T R
Z
Z
2
√
2
− T4 (E−E 0 )2
0
e
≤ e π |f (E)|
µ(dE ) µ(dE) .
(2.40)
For T > 1, the integral in brackets is estimated using UαH of µψ
∞ Z
∞
X
2
2C X − n2
− T4 (E−E 0 )2
0
e
e 4
µ(dE ) ≤ α
n+1
n
T
0
n=0 T ≤|E−E |< T
n=0
which completes the proof.
Based on Strichartz’s theorem, Last [107] proved the following refinement of Corollary 2.3.
16
2. SOME MATHEMATICAL GROUNDWORK
Theorem 2.7 (Last) Let H be a self-ajoint operator on some Hilbert space
H and assume the spectral measure of ψ with respect to H is UαH for some
α. Then there is C < ∞ such that for all p ≥ 1 and T > 0, and compact
operators A with tr |A|2p < ∞:
p1
Z
C
1 T
−itH 2
2p
lim
Ae
ψ dt ≤
.
(2.41)
tr |A|
T →∞ T 0
Tα
Proof: We employ the canonical decomposition of compact opertors [128],
i.e, there P
exist an > 0 and orthornomal basis {φn } and {ρn } in H such
that A = ∞
n=0 an |ρn ihφn |.R Abbreviating the Césaro average of a function
T
1
g ∈ L by AvT (g) := T −1 0 g(t) dt, Hölder’s inequality yields
AvT kAψ(·)k
2
=
∞
X
a2n AvT |hφn , ψ(·)i|2
n=0
≤
∞
X
n=0
!1/p
a2p
n
∞
X
!1/q
AvT
q
|hφn , ψ(·)i|2
.
n=0
(2.42)
Strichartz’ theorem guarantees h|hφ, ψ(·)i|2 iT ≤ CT −α which yields the
following upper bound on the above sum:
q−1 X
∞
∞
X
C
2 q
2
AvT |hφn , ψ(·)i|
Av
≤
|hφ
,
ψ(·)i|
T
n
Tα
n=0
n=0
q−1
C
=
.
(2.43)
Tα
Inserting (q − 1)/q = 1/p completes the proof.
Lecture 3 (excl.
Strichatz/Last)
28.4.2010
3. Appendix: Herglotz functions
A holomorphic function F : C+ → C+ is called a Herglotz function. Such functions naturally arise as the Borel-Stieltjes transformation
of finite Borel measures µ on R:
Z
µ(dλ)
, z ∈ C+ .
(2.44)
F (z) :=
λ−z
The representation theorem for Herglotz functions reveals their relation.
NOTES
17
Proposition 2.8 The Borel-Stieltjes transform of any finite Borel measure µ is a Herglotz function satisfying
µ(R)
|F (z)| ≤
.
(2.45)
Im z
Moreover, the measure µ can be recovered from F using:
Z
1
1 λ2
(µ ((λ1 , λ2 )) + µ ([λ1 , λ2 ])) = lim
Im F (λ + iε) dλ . (2.46)
ε↓0 π λ
2
1
Conversely, if F is a Herglotz function satisfying
C
|F (z)| ≤
, z ∈ C+ ,
(2.47)
Im z
then there exists a unique Borel measure µ with µ(R) ≤ C and F its BorelStieltjes transformation.
A proof can be found in [146].
Notes
The material in the first section was taken from [128, 153]. The construction of the spectral measure using the representation theorem for Herglotz functions is discussed in detailed in [146, 41]. The latter also includes
more material on Herglotz functions.
The material on the relation of spectra and dynamics was taken from
[146] and [107]. Generalizations of such statements dealing with continuum models, H = L2 (Rd ) or α-continuous spectral measures can be found
in [26, 107]. Further relations concerning the speading of generalized eigenfunctions and dynamical properties have been obtained in [83].
CHAPTER 3
Modeling disorder
1. Stochastic processes
As will become clear in the course of this section, random potentials
are most naturally given in terms of ergodic stochastic processes on some
probability space. We first recall some basic notions and facts, again in a
somewhat informal manner. The reader is invited to consult the literature in
the notes, in particular [65, 68].
1.1. Basic notions. Any family (X(ξ))ξ∈I of random variables , X(ξ) :
Ω → R, ω 7→ X(ξ, ω), on some probability space (Ω, A, P) is called a
stochastic process with index set I. We will mainly be concerned with the
case I = Zd .
An example fitting this framework are random variables attached to
the sites of a lattice, ξ ∈ Zd . In this situation, the probability space is
d
d
canonically realized as Ω = RZ with the Borel sets A = B(RZ ) serving as
the σ-algebra. The latter is the smallest σ-algebra generated by the cylinder
sets
Z(Iξ1 , . . . , Iξn ) := {ω | ωξk ∈ Iξk for all k = 1, . . . , n} ,
with Iξk ⊂ R Borel. A celebrated theorem of Kolmogorov ensures that
any probability measure can be uniquely characterized by the marginals,
P (Z(Iξ1 , . . . , Iξn )), provided they satisfy some compatibility conditions. In
this setup, the family
X(ξ, ω) = ωξ ,
indexed by ξ ∈ Zd , then forms a stochastic process.
The case of independent and identically distributed (iid) random
variables corresponds to P given by the product of a Borel measure p on R.
Homogeneity of the situation is expressed through the family of shifts on
d
Ω = RZ defined by
(Sx ω)ξ := ωξ−x ,
19
ξ, x ∈ Zd .
(3.1)
20
3. MODELING DISORDER
They leave the probability of any cylinder set Z(Iξ1 , . . . , Iξn ) invariant :
P (Z(Iξ1 , . . . , Iξn )) =
n
Y
p(Iξj ) = P Sx−1 Z(Iξ1 , . . . , Iξn ) .
j=1
d
Since this property extends to any set in B(RZ ), the shifts are an example
of a family of measure preserving transformations, i.e.,
P(Sx−1 A) = P(A)
for all A ∈ A.
The stochastic process X(ξ, ω) = ωξ corresponding to iid random variables
is then identified to be homogeneous (or stationary).
Definition 3.1 A stochastic process (X(ξ))ξ∈Zd is called homogenous if
there exists a family of measure preserving transformations (Tx )x∈Zd such
that
X(ξ, Tx ω) = X(ξ − x, ω) .
(3.2)
Homogeneity is a physically plausible requirement. In the context
of disordered solids such as alloys it is the natural generalization of the
periodicity of the ideal cystal. Whereas the latter requires strict translation
invariance, the covariance (3.2) is a statement of translation invariance on
the average: shifted realizations are equally likely.
1.2. Ergodicity. However, in order to discover determinism in randomness, the stronger notion of ergodicity has proven to be vital.
Definition 3.2 A family of measure preserving transformations (Tx )x∈I on
a probability space (Ω, A, P) is called ergodic if any event A ∈ A which is
invariant, i.e., Tx−1 A = A for all x ∈ I, has probability zero or one.
The importance of ergodicity in connection with determinism is further exemplified in the following lemma.
Lemma 3.3 Any random variable, X : Ω → R ∪ {∞}, which is invariant
under a family (Tx )x∈I of ergodic transformations, i.e.,
X(Tx ω) = X(ω) ,
for all x ∈ I,
is almost surely constant, i.e., there exists c ∈ R ∪ {∞} such that
P (X = c) = 1 .
1. STOCHASTIC PROCESSES
21
Proof: The distribution function t 7→ P (X ≤ t) is monotone increasing
on (−∞, ∞] and due to invariance of X takes values in the set {0, 1} only.
A candidate for the constant thus is
c := inf {t ∈ (−∞, ∞] | P (X ≤ t) = 1} ,
which by construction satisfies
[
1
P (X < c) = P
X ≤c− n
= 0.
n∈N
This implies the assertion: P (X = c) ≥ P (X ≤ c) − P (X < c) = 1.
It is natural to refer to homogeneous stochastic processes whose measure preserving family is ergodic as ergodic themselves.
Definition 3.4 A stochastic process (X(ξ))ξ∈Zd is called ergodic if there
exists a family of ergodic transformations (Tx )x∈Zd such that (3.2) holds.
Example 3.5 The canonical realization of iid random variables, X(ξ, ω) =
ωξ , ξ ∈ Zd , together with the shifts (3.1). For a proof of this assertion we
use the fact that the shift operators are even mixing, i.e., for any pair of
events A, B ∈ A:
P A ∩ Sx−1 B −→ P(A) P(B) as |x| → ∞.
(3.3)
In our situation, this is easily checked for cylinder sets and hence applies
d
to all events in A = B(RZ ). If E ∈ A is invariant, then (3.3) implies
P(E)2 = P(E) and hence P(E) ∈ {0, 1}.
Another example fitting the framework are almost-periodic functions.
Example 3.6 If Ω = [0, 2π), A = B([0, 2π)) and P is uniform distribution
on the torus, then for any irrational α ∈ (0, 1)
X(ξ, ω) := cos(2πα ξ + ω) ,
ξ ∈ Z,
defines an ergodic stochastic process. The family of ergodic transformations is given by the irrational rotations, Tx ω := ω − 2πα x mod 2π. We
will leave the proof as an exercise, cf. [105].
One of the most important results in ergodic theory is Birkhoff’s ergodic theorem. It states that the law of large numbers applies to ergodic
stochastic processes when sampled over boxes ΛL := {x ∈ Zd | |x|∞ < L}.
22
3. MODELING DISORDER
Proposition 3.7 (Birkhoff) Let (X(ξ))ξ∈Zd be an ergodic stochastic process and suppose that X(0) ∈ L1 (Ω, P). Then P-almost surely:
1 X
lim
X(ξ) = E [X(0)] .
(3.4)
L→∞ |ΛL |
ξ∈Λ
L
We will encounter several applications of this result later on. A modern proof and more information can be found in [100].
2. Ergodic operators
2.1. Definitions. Our next focus will be operator-valued maps defined
on some probability space (Ω, A, P) which take values in the set of selfadjoint operators on a separable Hilbert space H. A first concern is their
measurability.
Definition 3.8 We will call maps ω 7→ H(ω) into to set of self-adjoint
operators weakly measurable if the functions ω 7→ hϕ , f (H(ω))ψi are
measurable for all f ∈ L∞ (R) and all ϕ, ψ ∈ H.
It suffices to check this property for all f (x) = (x − z)−1 with z ∈
C\R, cf. [25, 72]. We will not dwell on this property in the following.
Our main interest concerns the case in which the probability space
carries a family of ergodic transformations.
Definition 3.9 Let (Ω, A, P) be a probability space with a family (Tx )x∈I
of ergodic transformations. Any weakly measurable map ω 7→ H(ω) into
the self-adjoint operators on a separable Hilbert space H is called a family
of ergodic operators if H(Tx ω) is unitarily equivalent to H(ω).
To give an example we note that the shifts (Sx )x∈Zd defined in (3.1)
form a representation of the group of lattice translations on the probability
space. This group may also be realized by
(Ux ψ) (ξ) := ψ(ξ − x)
(3.5)
as unitary operators (Ux )x∈Zd on the Hilbert space `2 (Zd ).
For any any ergodic stochastic processes (V (x))x∈Zd the corresponding multiplication operator V (ω) on `2 (Zd ) transforms covariantly:
Ux V (ω) Ux−1 = V (Sx ω) .
(3.6)
Since the Laplacian ∆ is invariant under Ux , any random Schrödinger operator of the form
H(ω) = −∆ + V (ω)
(3.7)
2. ERGODIC OPERATORS
23
constitutes a family of ergodic operators on `2 (Zd ) for which
Ux H(ω) Ux−1 = H(Sx ω) .
(3.8)
Definition 3.10 Any family of ergodic operators H(ω) in `2 (Zd ) satisfying the covariance relation (3.8) with the translation operators Ux defined
in (3.5) will be called a standard ergodic operator.
An example for a non-standard ergodic operator are Schrödinger operators with magnetic fields. The simplest one is the Hofstädter operator
[59] perturbed by a random potential, i.e., an operator of the form
H(ω) = −∆A + V (ω)
acting in `2 (Z2 ), where the magnetic Laplacian ∆A corresponds to a constant magnetic field B ∈ [0, 2π):
(∆A ψ) (x) = e−iBx2 ψ(x1 + 1, x2 ) + eiBx2 ψ(x1 − 1, x2 )
+ ψ(x1 , x2 + 1) + ψ(x1 , x2 − 1) − 4 ψ(x) . (3.9)
Physically, −∆A generates the quantum dynamics of a particle on Z2 subject to a constant perpendicular magnetic field of strength B. Clearly, this
operator is not invariant under all Ux . We will return to its interesting spectral features in a later chapter.
2.2. Deterministic spectra. For ergodic operators certain questions have
a predictable answer. Examples are the spectra and the density of states.
The latter will be the topic of the next chapter. Concerning the spectra, the
following result dates back to L. Pastur [122] with extensions by H. Kunz,
B. Souillard [103] and W. Kirsch and F. Martinelli [72].
Theorem 3.11 (Pastur) The spectrum of a family of ergodic operators
H(ω) is P-almost surely non-random, i.e., there is Σ ⊂ R such that
P (σ(H) = Σ) = 1 .
(3.10)
The same applies to any subset in the Lebesgue-decomposition of the spectrum, i.e., there are Σ# ⊂ R, # = ac, sc, pp, such that
P σ # (H) = Σ# = 1 , # = ac, sc, pp .
The above non-random set Σ is called the almost-sure spectrum associated with H(ω) and likewise for the spectral components.
Let us stress that the set of eigenvalues of a family of ergodic operators is usually heavily dependent on the realization ω. Only the closure of
Lecture 4
30.4.10
24
3. MODELING DISORDER
this set is deterministic. Just think of the random multiplication operator
corresponding to iid random variables on Zd .
Proof of (3.12): For any E1 , E2 ∈ R, the functions
ω 7→ XE1 ,E2 (ω) := dim Range P(E1 ,E2 ) (H(ω)) ,
defined on the underlying probability space (Ω, A, P) are:
(1) measurable, i.e., random variables taking values in [0, ∞].
This is most easily seen by identifying
P them as (possibly divergent)
non-negative series, XE1 ,E2 (ω) = ∞
k=1 hψk , P(E1 ,E2 ) (H(ω)) ψk i,
given in terms of an arbitrary orthonormal basis (ψk ) in H.
(2) invariant under the action of the ergodic transformations (Tx )x∈I ,
dim Range P(E1 ,E2 ) (H(Tx ω)) = dim Range P(E1 ,E2 ) (H(ω)) ,
by unitary equivalence.
Lemma 3.3 then ensures that there are constants cE1 ,E2 ∈ [0, ∞] such that
P (XE1 ,E2 = cE1 ,E2 ) = 1. The characterization (2.17) of the spectrum as
the support of the spectral projections identifies
Σ = {E ∈ R | For all E1 , E2 ∈ Q with E1 < E < E2 :
cE1 ,E2 > 0}
as a possible candidate for the almost-sure spectrum. The reason for choosing intervals with rational endpoints becomes apparent from the fact that
the event
\
{XE1 ,E2 = cE1 ,E2 } ⊂ {σ(H) = Σ} ,
E1 ,E2 ∈Q
cE1 ,E2 >0
still has probability one, since it is a countable intersection of such events.
The same proof idea also applies to the components of the spectrum. The
only subtle point is the measurability of the spectral projections associated
to the restriction of H(ω) to the subspaces H# , # = ac, sc, pp. We refer
the interested reader to [72, 103].
2.3. Determining the spectrum. Determining the almost-sure spectral
components of an ergodic operator is in general a hard question which much
of the later chapters are devoted to. In contrast, the almost-sure spectrum is
much easier to identify. One strategy is based on the construction of Weyl
sequences, i.e., sequences of approximate normalized eigenfunctions.
2. ERGODIC OPERATORS
25
Proposition 3.12 (Weyl criterion) Let A : D(A) → H be a self-adjoint
operator on a Hilbert space H. Then
n
o
σ(A) = λ ∈ R | ∃ (ψn ) ⊂ D(A), kψn k = 1 : lim k(A − λ)ψn k = 0 .
n→∞
A more general statement and a proof can be found in [146].
Let us illustrate the strategy to determine the spectrum in case of a
standard ergodic operator of the form (3.7). Associated to the random potential (V (x))x∈Zd defined on some probability space (Ω, A, P) are two notions of the support of the probability distribution:
supp1 (P) := {λ ∈ R | ∀ε > 0 : P (|V (x) − λ| < ε) > 0}
supp2 (P) := λ ∈ R | ∀ε > 0 , Λ ⊂ Zd finite :
P sup |V (x) − λ| < ε > 0
x∈Λ
The first is simply the support of the distribution of the single random variable V (0). It is the relevant notion if one is interested in the closure of
the range of the random function x 7→ V (x, ω). It is left as an exercise
to the reader to show that for any ergodic process (V (x))x∈Zd one has the
almost-sure equality:
supp1 (P) = {V (x, ω) | x ∈ Zd } .
(3.11)
The support supp2 (P) is always contained in the first, and the inclusion is
strict if there is substantial correlation between sites. The second notion
is most relevant for determining the almost-sure spectrum with the help of
Weyl sequences. For λ ∈ supp2 (P) typical realizations of the random process exhibit arbitrarily large regions where x 7→ V (x, ω) is almost constant.
It is those regions which accommodate approximate eigenstates of energies
in the range λ + [0, 4d]. This summarizes the proof idea of the following
result by H. Kunz and B. Souillard [103] which was generalized in [73].
Theorem 3.13 (Kunz/Souillard) For the family H(ω) = −∆ + V (ω) of
standard ergodic operators on `2 (Zd )
[0, 4d] + supp2 (P) ⊂ σ (H(ω)) ⊂ [0, 4d] + supp1 (P) ,
(3.12)
for P-almost all ω.
Proof: The proof of the second inclusion is based on (2.13) in which we
set A = H(ω) and B = V (ω) + 2d such that kA − Bk ≤ 2d, cf. (2.10).
26
3. MODELING DISORDER
Since σ(B) = σ(V (ω)) + 2d, the assertion follows from
σ(V (ω)) = {V (x, ω) | x ∈ Zd } = supp1 (P) ,
cf. (3.11).
For a proof of the first inclusion, we pick λ ∈ σ(−∆) = [0, 4d] and a
corresponding Weyl sequence (ϕn ) ⊂ `2 (Zd ). In fact, by a suitable smooth
truncation we may even assume that ϕn has compact support. For each
n ∈ N we consider the event
1
d
Ωn := There is j ∈ Z : sup |V (x + j) − µ| < n .
x∈supp ϕn
If µ ∈ supp2 (P) these events have a non-zero probability and are invariant
under the shifts (Sx )x∈Zd . By ergodicity we conclude that they are almost
certain, P (Ωn ) = 1. Therefore their intersection
\
Ω0 :=
Ωn
n∈N
is almost certain too. By construction, for any ω ∈ Ω0 there is a sequence
(jn ) ⊂ Zd such that
ψn := ϕn (· − jn )
is a Weyl sequence for H(ω) and λ + µ ∈ σ(H(ω)). This follows from
k(H(ω) − λ − µ) ψn k ≤ k(−∆ − λ) ψn k + k(V (ω) − µ) ψn k
≤ k(−∆ − λ) ϕn k + sup |V (x + jn ) − µ| kϕn k ,
x∈supp ϕn
which goes to zero as n → ∞.
In case of iid random variables, it is an easy exercise to show that
supp1 (P) = supp2 (P). Therefore the almost-sure spectrum may be determined explicitly.
Corollary 3.14 In case (V (x))x∈Zd are iid with common distribution p(dv) :=
P (V (x) ∈ dv),
σ (H(ω)) = [0, 4d] + supp p
for P-almost all ω.
Lecture 5
5.5.10
Notes
Most of material of this chapter was taken from [65, 68].
We have skipped examples of stochastic processes with continuous
index set I = Rd . For such a notion of homogeneity and ergodicity may
NOTES
27
be defined by implementing either the full translation group or subgroup
of lattice translations on the probability space. Important examples of such
processes like Gaussian processes are discussed in [65, 25, 123]. In physics
such processes feature in models of amorphous material [110, 109].
The measurability issue is discussed in more detail in [25, 72].
Our definition of ergodic operators is slightly more general (and not
standard) in comparison to what is discussed there. In fact, our notion of
standard ergodic operators agree with the usual ergodic operators. In the
standard case, much more can be said: the range of the spectral projections
are almost surely either zero or infinity. Therefore the discrete spectrum is
empty. For details see [65, 25].
The last section is a baby version of a more general framework to
determine the spectra of ergodic operators. In fact, the local supremum
norm arising in the definition of supp2 (P) can be used to render the canond
ical choice of the probability space, Ω = RZ , into a polish space, i.e., a
complete metric space with a countable dense subset. The set supp2 (P) is
then identified as the set of constant functions within the support of P with
respect to this topology. For more information, see [65, 25, 73].
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