The ux-across-surfaces
theorem and its implications
for scattering theory
Dissertation
an der Fakultat fur Mathematik und Informatik
der Ludwig-Maximilians-Universitat Munchen
Stefan Teufel
March 1999
2
Contents
Introduction
Summary
1 Potential scattering
1.1
1.2
1.3
1.4
1.5
1.6
The fundamental dynamics . . . . . . . . . .
Denition of the scattering cross section . .
Time asymptotics of scattering states . . . .
The ux-across-surfaces theorem . . . . . . .
Initially free states and the S-matrix . . . .
Scattering of almost momentum eigenstates
2 Mathematical results
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2.1 Regularity of the generalized eigenfunctions . . . . . . . . . .
2.2 The ux-across-surfaces theorem . . . . . . . . . . . . . . . . .
2.2.1 The free ux-across-surfaces theorem . . . . . . . . . .
2.2.2 The ux-across-surfaces theorem for short-range potentials and wave functions without energy cutos . . .
2.3 Approximate eigenstates of W . . . . . . . . . . . . . . . . .
b k)j2 ! (k0) . . . . . . . . . . . . . . . . . . . .
2.4 The limit j(
2.5 Some technical lemmas . . . . . . . . . . . . . . . . . . . . . .
Bibliography
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4
Introduction
Scattering experiments are central to most of modern physics. Much of our
knowledge about the world is derived from such experiments. Prominent
examples date back to the discovery of the atomic nucleus by Rutherford.
Today the scattering of elementary particles in modern accelerators is almost
the only source of new experimental information about the world in the small.
Hence it should come as no surprise that the theoretical and mathematical aspects of scattering theory are also well established. Actually the
mathematical history of scattering theory has been as fruitful as the physical. For example, the elds of Schrodinger operators and spectral theory are
inseparably connected with scattering theory. One might therefore expect
concerning the subject of this work, namely potential scattering of a nonrelativistic, spinless quantum particle, that it has been covered to such an
extent that hardly anything new can be said about it.
However, this is true only if one sticks to the classical questions of mathematical scattering theory, e.g., that of asymptotic completeness of the wave
operators, analyticity of the scattering operator or numerical calculations
of scattering amplitudes. Surprisingly, the connection of all these abstract
objects to genuine physics, i.e., ultimately to real world experiments, has
long been neglected by mathematicians as well as by physicists: Abstract
scattering theory describes \idealized" situations, but it is not clear what
these situations are an idealization of.
The goal of this work is to present a derivation of scattering theory that
is both mathematically and physically complete. We will start with a well
dened underlying fundamental dynamics and end up with the usual formulas of abstract scattering theory. Hence it will become clear what the
usual formulas of scattering theory are approximations of and exactly which
idealizations are involved in the approximation.
While the presentation in Chapter 1 is mostly new, many of the ideas
and concepts involved are not: Daumer [10] (1995) (see also Daumer, Durr,
Goldstein and Zangh[11], 1996) discussed quantum scattering from the point
of view of Bohmian mechanics, a Galilei invariant theory for the motion of
5
point particles that is empirically equivalent to non-relativistic quantum mechanics. Grounding the analysis in a theory for which particles have trajectories allows for more precise arguments and for a clear focus on the relevant
questions. Therefore we will also invoke Bohmian mechanics as the basic
dynamics. However, it should be stressed that most parts of Chapter 1 are
relevant from an orthodox quantum mechanical perspective as well.
This is true, in particular, for the so-called ux-across-surfaces theorem
that was conjectured by Combes, Newton and Shtokhamer [9] in 1975 as
a quantum mechanical theorem. Its fundamental importance to scattering
theory became obvious in [10] and [11], however, only in connection with
Bohmian mechanics. Nonetheless, numerous standard proofs of the uxacross-surfaces theorem for dierent kind of potentials and dierent assumptions on the initial wave function have been provided. Amrein and Zuleta
[5] gave the rst proof for the short-range case in 1997, while Amrein and
Pearson [3] then covered a large class of long-range potentials also in 1997.
However, both proofs require technical assumptions called energy-cutos that
limit the applicability of the theorem. In the present work we will cure this
problem for some short-range potentials. Recently DellAntonio, Panati and
Teta1 proved the ux-across-surfaces theorem for so-called point interactions,
corresponding, heuristically, to a -potential, using methods similar to ours.
The mathematical problems encountered in the analysis of Chapter 1,
to the extent that they are not resolved there, will be treated in Chapter
2. There we rely, of course, on many existing results from the vast eld of
mathematical scattering theory. When appropriate, e.g., standard methods
from so-called time-dependent scattering theory are applied (see Section 2.3).
It turns out, however, that, from a mathematical point of view, some of the
new questions that arise in Chapter 1 are quite dierent from those classical
problems of mathematical scattering theory mentioned above. The main
mathematical novelty of theorems like the ux-across-surfaces theorem (see
Section 2.2) is that such a problem cannot naturally be formulated as a
\Hilbert space problem." In order to obtain satisfying answers we develop
new methods and alter existing ones.
The approach we chose in order to prove the ux-across-surfaces theorem
involves the well known generalized eigenfunctions (x; k). As the plane
waves eikx are \eigenfunctions" of the free Hamiltonian H0 = (1=2), the
functions (x; k) are \eigenfunctions" of the Hamiltonian H = H0 + V .
We will need, in particular, regularity of the generalized eigenfunctions as
functions of k, the \momentum" variable. Since in the literature there are
no sucient results available, we prove a theorem about the regularity of
1 Private communications.
6
the generalized eigenfunctions in Section 2.1. This theorem turns out to be
relevant and useful even beyond its applications in this work.
This work contains the rst completely rigorous derivation of the standard formulas of quantum scattering theory. However, although there are
no mathematical gaps in our derivation, some of the important results are
expected to hold with much greater generality than we are able to establish.
Our whole mathematical discussion is concerned only with short-range potentials, thereby excluding the physically important Coulomb potential. It
seems heuristically reasonable to conjecture, for example, that the fundamental ux-across-surfaces theorem holds whenever asymptotic completeness of
the (modied) wave operators holds. Hence, we believe that the theorems
formulated in this work answer the physical questions in principle and set
a proper framework for future work, but they need to be proven in greater
generality. This could turn out to be a demanding task, however, since, as we
shall see, the techniques that have proven useful in the treatment of asymptotic completeness involve some major drawbacks in the present framework.
7
8
Summary
The rst part of this work is concerned with potential scattering of a nonrelativistic, spinless particle moving according to the laws of Bohmian mechanics, respectively quantum mechanics. The standard formulas of abstract
scattering theory for the scattering cross section are derived from rst principles. Although the discussion in Chapter 1 will be based on mathematically
precise statements, the main focus will be on the physical aspects of the
problems under consideration.
In Section 1.1 we start by introducing Bohmian mechanics and quantum
mechanics as the fundamental dynamics for which the corresponding scattering theory is derived later on. The relevant results from the existence theory
for these dynamics are quoted.
The analysis of scattering theory begins in Section 1.2, where we dene
the scattering cross section, the basic object that connects scattering theory
with experiment. In analogy to classical mechanics the scattering cross section is dened as a probability measure on the unit sphere. Assume that the
detectors in a scattering experiment cover a spherical surface with radius R
containing the scattering center. One is interested in the probability that the
particle crosses, at some random time during the experiment, some subset
R := fx 2 IR3 : x = R!; ! 2 g of this sphere, being a subset of the
unit sphere, or, hopefully equivalent, that a detector covering this surface is
triggered. For large R these probabilities should be independent of R and
dene a probability measure on the unit sphere, the scattering cross section
measure. Since at this point the concept of the probability for a particle to
cross a surface appears, the relevant analysis can be done rigorously only for
Bohmian mechanics (or for some other formulation of quantum mechanics,
like stochastic mechanics, that seriously incorporates trajectories)|unless
one also analyzes the detectors quantum mechanically, a complication best
avoided and, indeed, rarely considered in scattering theory. However, the formulas one ends up with, make sense, at least heuristically, also for orthodox
quantum mechanics.
The rest of Chapter 1 is concerned with the computation of the scatter9
ing cross section measure, that is, with recovering from its basic denition
the standard formulas of scattering theory. As a rst step, in Section 1.3,
we briey discuss the time asymptotics of scattering states and review some
important results that usually come under the name of asymptotic completeness.
Then a fundamental theorem, rst conjectured by Combes, Newton and
Shtokhamer [9] in 1975, is motivated and explained. The quantum probability ux of a particle integrated over time and a distant surface gives the
probability for the particle crossing that surface at some time. The relation
between these crossing probabilities and the usual formula for the scattering
cross section in terms of Fourier transforms is provided by the ux-acrosssurfaces theorem.
The two remaining sections of Chapter 1 are concerned with the idealizations necessary to make contact with the standard formulas of \naive"
time-independent scattering theory. In Section 1.5 the so-called scattering
operator or S -matrix is introduced. We shall argue that under suitable circumstances the wave operator appearing in the ux-across-surfaces theorem
can be replaced by the (in some sense simpler) scattering operator.
Finally, in Section 1.6, we aim at an expression for the scattering cross
section independent of the precise form of the initial wave function, and
sensitive only to the (small) support of its Fourier transform. Thereby we
recover the standard formula for the scattering cross section that has been
conrmed in thousands of experiments.
In Chapter 2 we formulate and prove theorems that solve the problems
and answer the questions encountered in Chapter 1. In Section 2.1 we rst
introduce the so-called generalized eigenfunctions as a tool that we use frequently in the rest of this chapter. We establish a theorem concerning the
regularity of the generalized eigenfunctions as functions of the momentum
variable. The results so obtained are completely new, even qualitatively, and
open up various possible applications beyond those in this work. The theorem connects the spatial decay of the potential V with the smoothness of
the generalized eigenfunctions (x; k) as functions of k, where x; k 2 IR3
and (x; k) are solutions of
1
2
2 x + V (x) (x; k) = jkj (x; k) :
Furthermore the theorem provides bounds on (x; ) and its partial derivatives with respect to k that are uniform in x. The usefulness of this result is
documented in two corollaries.
In Section 2.2 we come to the main result of this work, the proof of
the ux-across-surfaces theorem. Although there are already two existing
10
proofs, we will rst provide a much simpler and shorter proof for the free
case, i.e., the case with vanishing potential. The ux-across-surfaces theorem
including a potential was also proved recently (see [5] and [3]), but only
for a class of wave functions that is restricted in an essential way by the
requirement of so-called energy cutos. We will establish the ux-acrosssurfaces theorem for a certain class of short-range potentials|essentially we
demand that jV (x)j = O(jxj 4 ) for some > 0 and jxj ! 1|but, and this
is new, for wave functions without energy cutos. The proof relies heavily
on the results established in Section 2.1.
Standard methods from time-dependent scattering theory are applied in
Section 2.3. We show that one can generate approximate eigenstates of the
wave operators W by moving wave functions (x) with Fourier transforms
b k) supported in a half-space P := fk 2 IR3 : k 0g in the direction
(
of towards spatial innity. This result justies the replacement made in
Section 1.5.
In Section 2.4 we use again the regularity of the generalized eigenfunctions
b k)j2 ! (k0) in the
established in Section 2.1 in order to perform the limit j(
formula for the scattering cross section measure. This is done, as explained
in Section 1.6, to obtain the usual formulas of quantum scattering theory
that depend only on k0, but not on further details of the wave function itself.
We conclude the chapter with a section containing two technical lemmas
used in the proof in Section 2.1. They have been moved to this point because
the proofs involve simple but lengthy calculations that would have obscured
the structure of the rest of the proof in Section 2.1.
11
Acknowledgements
It is a pleasure to thank Prof. Dr. Detlef Durr for providing such an interesting and fruitful
topic for my Ph.D. thesis and doing such a great job as its advisor.
However, there is much more: to him and to Prof. Dr. Sheldon Goldstein, who invited
me for nine months to Rutgers, I owe many thanks for their great personal engagement
and continuous promotion in all matters, for their trust in my abilities, for the numerous
interesting and deep things I learned from both of them and, last but not least, for the
great time I had working with them in Munich and in Rutgers.
Next I would like to thank Dr. Karin Munch-Berndl for numerous discussions, for
carefully checking great parts of the proofs in this work and for the good overall teamwork.
I'm also very greatful to Prof. Dr. Nino Zangh, who helped me in many discussion to better
understand, in particular, the physicist's view of scattering theory and to arrive at the
well structured picture this work hopefully reects.
Furthermore I would like to thank
Prof. Dr. Herbert Spohn for being co-advisor of this work, for valuable discussions
and stimulating questions;
Prof. Dr. Joel Lebowitz, Prof. Dr. Micheal Kiessling and the rest of the math
department of Rutgers University for their kind hospitality;
Prof. Dr. Werner Amrein for sending me his papers [5] and [3] prior to publication
and for pointing out reference [20];
Prof. Dr. Avraham Soer for interesting discussions.
There are of course many others who helped me in many ways and I want to express
my sincere thanks to all of them, especially to my parents and my friends.
I am also grateful for the nancial support I got directly from DFG and as a member
of the Graduiertenkolleg \Mathematik im Bereich ihrer Wechselwirkung mit der Physik".
12
Chapter 1
Potential scattering of a
Bohmian or quantum particle
1.1 The fundamental dynamics
In this section we introduce the fundamental dynamics for which we will derive the corresponding scattering theory later on. While the more precise line
of thought will start with particles moving according to Bohmian mechanics,
one can also think of quantum mechanical \particles". We will see, however,
that in the case where particles have positions, i.e., in the Bohmian case, the
denition of a scattering cross section is straight forward, while in the case of
quantum mechanics one has to rely on heuristic arguments and on physical
intuition.
Bohmian mechanics (see e.g. [8], [15]) is a Galilean invariant theory for
the motion of point particles. Consider a system of N particles with masses
m1 ; : : : ; mN and potential V = V (Q1 ; : : : ; QN ), where Qk 2 IR3 denotes the
position of the kth particle. The relevant conguration space is an open
subset of IR3N , for example the complement of the set of singularities of V ,
and shall be denoted by . The state of the N -particle system is given by
the conguration Q = (Q1 ; : : : ; QN ) 2 and the Schrodinger wave function
on conguration space . On the subset of where the wave function
6= 0 and is dierentiable, it generates a velocity eld v = (v1; : : : ; vN ),
k (q )
vk(q) = mh Im r(
q) ;
k
(1.1)
the integral curves of which are the trajectories of the particles. Thus the time
evolution of the state (Qt ; t) is given by a rst-order ordinary dierential
13
14
CHAPTER 1. POTENTIAL SCATTERING
equation for the conguration Qt ,
dQt = vt (Q ) ;
t
dt
and Schrodinger's equation for the wave function t,
!
2
N h
X
@
t (q )
ih @t =
k + V (q) t(q) ;
2
m
k
k=1
(1.2)
(1.3)
where rk and k denote the gradient and the Laplace operator in IR3 and
the potential V is a real-valued function on .
In orthodox quantum mechanics the time evolution of the state t is given
by a one-parameter unitary group Ut on a Hilbert space H. Ut is generated
by a self-adjoint operator H , which on smooth wave functions in H = L2 (
)
is given by
2
N h
X
(1.4)
H=
2m k + V =: H0 + V ;
k=1
k
i.e., Schrodinger's equation is regarded as the \generator equation" for Ut.
The connection to physics is given by Born's statistical interpretation of the
wave function: the probability for nding a system at time t in a certain
region of conguration space is given by
IP(Q 2 ) =
Z
jt(q)j2 dq :
(1.5)
Remarkably it turns out that Bohmian mechanics and usual quantum mechanics make the same empirical predictions, provided the Bohmian systems
one looks at are in \quantum equilibrium"[15]: The dynamical system dened by Bohmian mechanics is associated with a natural measure, given by
the density j0j2 on
R conguration space . If 0 is normalized, i.e., the
L2 -norm k0k = ( j0j2 dq) 21 = 1, then the density j0j2 denes a probability measure IP on conguration space that plays the role usually played
by the \equilibrium measure". Assuming the existence of the dynamics Qt
for congurations|a result we shall comment on later|the density jtj2 is
equivariant [15]:
0 = j0j2 ! t = jtj2 for all t 2 IR ;
(1.6)
where t denotes the image density of 0 under the motion Qt. This follows
from comparing the continuity equation for an ensemble density t (q)
N
@t (q) + X
rk [vkt (q)t (q)] = 0
@t
k=1
(1.7)
1.1. THE FUNDAMENTAL DYNAMICS
15
with the quantum continuity equation, an identity for classical solutions of
Schrodinger's equation,
N
@ jt (q)j2 + X
rk jkt (q) = 0 ;
@t
k=1
(1.8)
where the quantum probability current j = (j1; : : : ; jN ) is given by
jk = vkjj2 = mh Im(rk ):
k
(1.9)
Therefore we say that an ensemble of Bohmian systems, each with the same
initial wave function 0, is in \quantum equilibrium", if the congurations
are distributed among the ensemble according to the density 0 = j0j2.
Then the phenomenological equivalence of Bohmian mechanics and quantum
mechanics for \position measurements" follows from equivariance, i.e., t =
jt j2 .
Now we come to the question of existence of the dynamics. In the case
of quantum mechanics the basic existence theorem and Stone's theorem (see
e.g. [24] Theorems VII.7 and VII.8) tell us that H generates a strongly continuous one-parameter unitary group Ut = e iHt such that t := Ut 0 satises
Schrodingers equation in the L2 sense whenever 0 2 D(H ), if and only if H
is self-adjoint. Thus the question of having a well dened time evolution for
every 0 2 L2 (
) is reduced to showing that the Hamiltonian operator H is
self-adjoint. Compatibility with the physical interpretation of jtj2 as a probability density is assured by unitarity, since then kt k = kUt 0k = k0k = 1.
For the Hamiltonians that we will consider later on standard results are
available, which settle the question of self-adjointness completely. We will
comment on that when we specify V more precisely.
In the case of Bohmian mechanics the question of existence of solutions
global in time is much more subtle. As far as Schrodinger's equation is concerned one naturally applies the mathematics developed for quantum mechanics. However, this is only one part of the dynamics. Even for smooth
wave functions the vector eld (1.1) will in general be singular at the nodes
of the wave function, i.e., whenever t(q) = 0. Also, in a nonrelativistic
theory where velocities are not bounded, it might happen that particles can
run o to innity in nite time.1 None the less, global existence for Bohmian
mechanics has been proven in great generality by Berndl et al. (see Corollary
3.2 in [7]):
1 For an example for this kind of behavior in Newtonian mechanics see e.g. [28].
CHAPTER 1. POTENTIAL SCATTERING
16
Theorem 1.1. Assume that
(i) the potential V = V1 + V2 is a C 1-function on where @ is a nite
union of (3N 3)-dimensional hyperplanes, V1 is bounded below and
V2 is H0-form bounded with relative bound a < 1;
(ii) the Hamiltonian H is the self-adjoint operator associated with the form
sum H0 + V ;
(iii) the initial wave function 0 is a C 1-vector of H , 0 2 C 1(H ), and is
normalized, k0k = 1.
Then Bohmian mechanics (that is, the coupled equations (1.2) and (1.3)) has
unique solutions global in time for IP-almost all initial conguration.
Remark 1.2.
Note that the condition on @ ts well with the 3dimensionality of physical space, since it allows for point-singularities in pair
potentials.
Remark 1.3.
The Hamiltonian H in Theorem 1.1 is dened via the
form sum of H0 + V . In this work all potentials V are assumed to be, in
particular, H0-operator bounded and H will be dened as the operator sum
H0 + V . Note that in this case both denitions coincide and Theorem 1.1
applies when replacing in (i) H0-form bounded by H0-operator bounded and
in (ii) form sum by operator sum.
Remark 1.4.
T
Recall that the set of C 1-vectors of H is dened by
= n=1 D(H n), where D(H n) is the domain of H n, and is dense in
L2 . Some special C 1-vectors are, for example, eigenfunctions and vectors
2 Ran(EI (H )), where EI (H ) is the spectral projection of H on a nite
energy interval I IR.
C 1(H )
1
After these general consideration we will develop scattering theory for
Bohmian respectively quantum mechanics.
1.2 The experimental situation and the denition of the scattering cross section
Roughly speaking, when talking about a scattering process one has the following general picture in mind: The initial state of a physical system is such
1.2. DEFINITION OF THE SCATTERING CROSS SECTION
17
that dierent parts of the system are spatially separated and evolve independently of each other. During time evolution they get closer and an interaction
takes place. After the interaction, however, the system splits up again into
dierent parts that evolve independently and separate spatially.
In the following we will concentrate on the case of scattering of two particles that evolve according to Bohmian mechanics. They interact via a potential that depends only on the spatial separation of the two particles, i.e.,
V (q) = V (q1 q2), and approaches zero if the spatial separation becomes
large, i.e., limjq1 q2j!1 V (q) = 0. For pair potentials the rst condition is
a direct consequence of the demand for translation invariance. And only
if the second condition holds we can expect that the two particles move
independently whenever they are suciently separated. In the case of translation invariant pair potentials, however, a further simplication of the problem is possible: after a coordinate transformation into relative coordinates
qrel = q2 q1 and center-of-mass coordinates qcom = (m1 q1 + m2q2 )=(m1 + m2 )
one is left, at least in the classical case, with two completely independent
motions. The center-of-mass moves according to the free dynamics, i.e., it
doesn't feel the potential, while the relative coordinates behave like those
of a single particle having the reduced mass m = (m1 + m2 )=(m1m2 ) and
moving in an external potential V (qrel). In the quantum mechanical case
this splitting is not entirely trivial, since although there is no interaction
that couples the motions of qrel and qcom there might still be entanglement
in the wave function. For a more detailed discussion of this splitting in the
quantum case see e.g. [2]. We won't discuss this question here any further
and take for granted that the only interesting part of the problem lies in
the motion of qrel. Therefore we will now focus on potential scattering of a
single Bohmian respectively quantum particle. From now on conguration
space will be physical space IR3 and we change notation from q and Q for
the generic conguration space variable and the actual conguration of the
Bohmian system to x and X , which are now vectors in IR3.
The basic object that connects abstract scattering theory with experiment
is the scattering cross section. The denition of the scattering cross section
for Bohmian particles was discussed in [10] in much detail. In the following we
will only repeat the main arguments leading to this denition. For quantum
mechanics the corresponding denition was rst given in [9]. However, in
that case the argument is less clear and only heuristic. The dierences are
discussed in [10].
In a typical scattering experiment the initial wave function 0 of a oneparticle system is prepared at time t = 0 and the particle moves together with
CHAPTER 1. POTENTIAL SCATTERING
18
the wave packet t towards the region where the potential V (x) is localized2.
After being scattered the particle leaves the inuence of the potential again
and is eventually detected by some experimental setup far away from the
scattering region. Since we do not want to take the detailed mechanism
of detection into account, we assume that our detectors are covering some
closed surface surrounding the scattering center and can measure where
the particle is crossing this surface without disturbing its motion prior to
detection.
If quantum equilibrium holds the initial position of the particle will be
random and therefore in general also the crossing position (and time) of
the particle will be random. Assuming that every particle starts inside the
region enclosed by the surface and leaves it eventually, the experimental
results provide us with a probability distribution for (rst) exit positions on
.
In order to be more specic we will assume from now on that the surfaces
are spheres with radius R centered at the origin, R := fx 2 IR3 : jxj =
Rg. A subset 1 of the unit sphere spans a cone that we denote by
C := fx 2 IR3 : x 2 ; 0g and its intersection with R is denoted by
R := C \ R = fRx 2 IR3 : x 2 g.
Lets assume that the initial wave function 0 is supported in some ball
BR with radius R and is such that for almost all initial positions the particle eventually leaves BR . Let Xe be the random variable3 on (IR3; IP0 )
that maps every initial position to the position where the particle rst hits
R . Assuming that the detectors in an experiment measure indeed this rst
exit position Xe,4 the object of experimental interest is the distribution of
this random variable IP0 (Xe 2 R), R R . Since in a typical scattering experiment the detectors are placed far away from the scattering center
where the particle moves freely and, presumably, on a straight line, we expect
this distribution to become independent of R when R gets suciently large.
Thus we dene the so-called scattering cross section as the probability
distribution on 1 given by
IP0 (Xe 2 R) :
(1.10)
0 () = Rlim
!1
Remark 1.5.
Note, that this denition is independent of the choices
for the origins of time and space. This is because the particle cannot reach
2 The fact that a Bohmian particle moves along with the guiding wave packet is a simple
consequence of equivariance.
3 For a rigorous denition and proofs of measurability see [10] and [6].
4 This assumption of course has to be justied by a careful analysis of the detailed
mechanism of detection.
1.2. DEFINITION OF THE SCATTERING CROSS SECTION
19
innity in nite time and because we assumed that it moves asymptotically
on a straight line. For a real experiment with detectors at nite distances
this translates into the fact that the results do not depend on small shifts in
the origins of time and space, where small means small with respect to the
time it takes the particles to reach the detector, respectively small compared
to R. Or put dierently, when the detectors are very far away, it doesn't
matter when they are turned on to start counting as long as they are turned
on before the rst particle can arrive. Thus we expect that if we compute
0 we end up with expressions independent of the origins of time and space.
In particular 0 = t , t = e iHt 0 should hold for all t < 1.
As a next step we will use a result established in [6] in order to cast
(1.10) into a more accessible form. For a Bohmian particle with initial wave
of crossings through a surface function 0 the expected total number Ntot
is given by
Z
Z
) = 1 dt jj t d j
IE0 (Ntot
(1.11)
0
by (assuming that IE0 (N )
and the expected number of signed crossing Nsgn
tot
is nite)
Z
Z
) = 1 dt j t d :
(1.12)
IE0 (Nsgn
0
outward
For Ntot every crossing of the surface is counted with 1, while for Nsgn
crossings are counted with 1 and inward crossings with 1. IE0 denotes the
expectation with respect to the quantum equilibrium measure IP0 on the
initial conditions and d is the normal surface element of . The number of
+ N ),
outward, respectively inward, crossings is then given by N+ = 21 (Ntot
sgn
1
respectively N = 2 (Ntot Nsgn).
In the scattering regime we expect that for large R the surface R is
crossed exactly once and only outwards. Hence for R large enough the expectation becomes a probability:
R ) IE0 (N R ) :
IP0 (Xe 2 R) IE0 (N+R ) IE0 (Ntot
sgn
(1.13)
Thus if in (1.13) equality holds in the limit R ! 1 the scattering cross
section measure is given by
0 () = Rlim
IP0 (Xe 2 R) = Rlim
!1 Z
!1
1 Z
= Rlim
dt j t d :
!1
0
R
Z1 Z
dt jj t dj
0
R
(1.14)
20
CHAPTER 1. POTENTIAL SCATTERING
Remark 1.6.
R ) =
The fact that for scattering states limR!1 IE0 (Ntot
R ) holds is part of the statement of the ux-across-surfaces
limR!1 IE0 (Nsgn
theorem (see Sections 1.4 and 2.2). From that also limR!1 IP0 (Xe 2 R) =
limR!1 IE0 (N+R ) follows.
We arrived at (1.14) under very general assumptions on the behavior
of scattered particles in Bohmian mechanics. The derivation was straight
forward and we haven't encountered any physical or mathematical problems.
Combes, Newton and Shtokhamer [9] arrived at the same formula for the
scattering cross section starting only with quantum mechanics as the basic
dynamics. In that case the argument is less clear, since without particle
trajectories the concept of the probability for crossing some surface is not
well dened. The quantum continuity equation (1.8) suggests of course to
interpret the expression
Z t2 Z
dt j t d
(1.15)
t1
as some sort of crossing probability also in quantum mechanics. But, since
it can be negative, its precise meaning is not clear at all. Only if we add the
particle trajectories the interpretation becomes straight forward.
From now on our goal will be to understand for which potentials V and
initial wave functions 0 our assumptions leading to the expression (1.14)
were justied and given by (1.14) is well dened and provides us indeed
with the probabilities of interest. Furthermore we have to connect (1.14)
to the usual formulas of quantum mechanical scattering theory, which have
been proven correct in experiment.
1.3 Time asymptotics of scattering states and
the wave operators
In order to evaluate the formula for the scattering cross section (1.14) we rst
have to understand the behavior of wave functions in the scattering regime.
Concerning this question many very general results have been established in
quantum mechanical scattering theory in the context of \asymptotic completeness". Here we shall only mention those results which will turn out to
be important to us in the further development. Many results can be shown to
hold in much greater generality by a variety of dierent methods (see Remark
1.7 for details and references).
For scattering potentials V vanishing suciently fast at spatial innity we
expect that all initial wave functions 0 corresponding to scattering states
will eventually move according to the free dynamics generated by H0. More
1.3. TIME ASYMPTOTICS OF SCATTERING STATES
21
precisely, for every scattering state 0 there should be a state 2 L2 (IR3)
such that
lim ke iHt0 e iH0 t k = 0 :
(1.16)
t!1
Thus one is interested in the existence and the range of the wave operator
iHt
W+ := s tlim
!1 e e
iH0 t :
(1.17)
If the wave operator exists, every state in its range eventually moves freely
in the sense of (1.16), since W+ maps every \free state" to the corresponding \scattering state" 0.5 The program of asymptotic completeness amounts to showing that W+ exists and that its range consists of
all states in L2 (IR3) orthogonal to the set of bound states of H . The set
of bound states of H is spanned by its eigenvectors and is thus given by
Hpp (H ), the spectral subspace of H spanned by the vectors generating a
pure point spectral measure. Since for self adjoint H the spectral decomposition H = Hpp (H ) Hcont(H ) holds, where Hcont (H ) is the spectral subspace
of H spanned by the vectors generating a continuous spectral measure, we
expect that Ran(W+) = Hcont (H ).
Naturally one can go through the same considerations for the behavior
of wave functions not in the far future but also in the far past and dene
analogously the wave operator
W := s
lim eiHt e
t! 1
iH0 t :
(1.18)
Whenever the wave operators exist, they obey the so-called intertwining
relations, which follow from a simple calculation:
e iHt W = We iH0 t :
(1.19)
And thus, on D(H0), we have by dierentiation,
HW = WH0 :
(1.20)
As a consequence of this relation and the fact that W are partial isometries one concludes (see e.g. [22] Theorem 4.3) that the restrictions of H to
Ran(W) are unitarily equivalent to H0 and, in particular, that Ran(W) Hac (H ), the absolutely continuous subspace of H . Therefore the question of
asymptotic completeness is deeply related to spectral theory. Knowing that
5 At rst sight it might seem more natural to dene the wave operator as the inverse
of W+ , i.e., as a map from states 0 to the corresponding future asymptotics . Then,
however, already the question of existence becomes highly nontrivial, because it is not
obvious what the domain should be.
22
CHAPTER 1. POTENTIAL SCATTERING
a scattering Hamiltonian H is asymptotically complete immediately implies
that it has no singular continuous spectrum.
We repeat some results on asymptotic completeness. Assume that the
potential V satises the following conditions denoted by (V)n, n 2:
(V)n V : IR3 ! IR and
(i) V is locally Holder continuous except at a nite number of
singularities.6
(ii) V 2 L2loc(IR3).
(iii) jV (x)j = O(jxj n ) for jxj ! 1 and some > 0, whenever
n is nite. If n = 1, then jV (x)j = O(jxj m) for all m 2 IN.
For n = 2 these are the conditions of Ikebe [18], under whichR H = H0 + V is
b k)j2 dk <
self-adjoint on the domain D(H ) = D(H0) = f 2 L2(IR3) : jk2(
1g, where b denotes the Fourier transform of . Thus for any 0 2 L2 (IR3 )
the unitary quantum mechanical time evolution t = eiHt 0 is well dened
for all t. According to Theorem 1.1 we have to further demand that V is
smooth except at its singularities in order to guarantee global existence also
of the Bohmian dynamics. However, since we would like to evaluate (1.14) in
as much generality as possible, and since it is an expression solely in terms
of the wave function, we will assume only the weaker quantum mechanical
conditions in the following.
Ikebe [18] showed that for potentials V satisfying (V)2 the wave operators of H = H0 + V dened on D(H0) are asymptotically complete. In
particular, Ran(W) = Hcont (H ) = Hac (H ) = [0; 1) and, furthermore, H
has no positive eigenvalues.
Remark 1.7.
Asymptotic completeness has been shown for a much
larger class of potentials by a variety of dierent methods. We introduced
the conditions (V)n, n 2, which are quite strong from the point of view
of asymptotic completeness, because they are appropriate for what is going
to follow. Some remarks concerning other potentials are, however, in order.
Most important, one has to distinguish short-range potentials with V (x) =
O(jxj 1 ) for jxj ! 1 and some > 0, and long-range potentials, as for
example the Coulomb potential V (x) = jxj 1 , for which the short-range
condition does not hold. The wave operators, as we introduced them, only
exist in the short-range case, since the motion under the inuence of a longrange potential never approaches that of a free particle (hence the name
6 V : D ! IR is locally Holder continuous if for every x 2 D there is an open neighborhood Ux D and an x > 0, cx > 0 such that jV (x) V (y)j cx jx yjx for all
y 2 Ux.
1.4. THE FLUX-ACROSS-SURFACES THEOREM
23
long-range). One can, however, introduce a modied free dynamics in order
to show existence of the wave operators with respect to this changed free
dynamics. Asymptotic completeness of the modied wave operators has also
been proven in great generality. Last but not least, N -particle scattering has
been considered too, where asymptotic completeness has been established
not too long ago. For modern references and developments in connection
with asymptotic completeness see, for example, the new book by Derezinski
and Gerard [13].
Hence, concluding this section, we know that for (V)2 -potentials all initial
states 0 orthogonal to all bound states propagate eventually like free states
and thus, presumably, exhibit the behavior we assumed for scattering states
in Section 1.2.
1.4 The ux-across-surfaces theorem
Now we come back to our original problem, namely to the evaluation of
(1.14). In order to get an idea what we can expect, we start with the simplest
case, the free time evolution where V = 0. The free time evolution is simple
because H0 is diagonalized by the Fourier transformation
2
(H0)(x) = (F 1 k F )(x) ;
(1.21)
2
where F denotes the Fourier transformation dened as an unitary map from
L2 (IR3 ) to itself. To distinguish functions in Ran(F ) from those in the original Hilbert space we write k for the variable of functions in the former.
Hence by k2=2 in (1.21) the operator of multiplication by the function k2=2
is understood. Note that in (1.21) we assumed, as we will do in the rest of
this work, that the physical constants h = m = 1. This is done to simplify
notation and means no loss of generality, because all results in the following
will be independent of the strength of the coupling. More precisely, we will
only refer to results that hold for whole families cV (x) of potentials, where
c is an arbitrary positive constant.7
By the functional calculus for self-adjoint operators (1.21) provides us
also with an explicit representation for e iH0t and thus for the Schrodinger
time evolution (assume b 0 2 L1 (IR3 )):
Z
ik2 t
1
iH
t
1
ikx i k2 t b
0
2
t (x) = (e 0)(x) = (F e F 0)(x) =
3 e e 2 0 (k) dk :
(2) 2
(1.22)
7 Sometimes, however, we have to exclude a discrete set of coupling constants c for
technical reasons.
CHAPTER 1. POTENTIAL SCATTERING
R
By replacing b 0(k) = (2) 23 e iky 0(y) dy (assume also 0 2 L1 (IR3 ))
24
in (1.22), then changing the order of integration and nally evaluating the
k-integration one arrives at
Z ijx yj2
1
t (x) =
e 2t 0(y) dy :
(1.23)
(2it) 32
As a direct consequence we can see already that a free particle
leaves any
R
bounded region G almost surely, since IP(X (t) 2 G) = G jt(x)j2 dx =
O(t 3). Using
e
ijx yj2
2t
=e
ijxj2
2t
e
ixy
t
+e
ijxj2
2t
e
ixy
t
e
ijyj2
2t
1
one can nally rewrite (1.23) as
ijxj2
ijxj2 Z
ijyj2 2t b x
2t
ixy
e
e
t
t(x) = 3 0( ) +
e
e 2t 1 0(y) dy : (1.24)
(it) 2 t (2it) 32
The fact that the second term in the right hand side of (1.24) vanishes for
t ! 1 in the L2 -sense is known as Dollard's lemma (for the simple proof see
e.g. [22] Theorem 3.1). Thus we have
ijxj2
e 2t b ( x ) = 0 :
lim
(
x
)
(1.25)
t
t!1 (it) 23 0 t By substituting k := xt and integrating only over some cone C , Dollard's
lemma immediately implies the so-called free \scattering-into-cones" theorem
lim
t!1
Z
Z
j
t(x)j2 dx = jb 0(k)j2 dk :
C
C
(1.26)
It says that the probability for nding a particle at very large times in some
cone C is equal to the L2 -weight of b 0 in C , which is in orthodox quantum
mechanics taken to be the probability for the particle's momentum being in
that cone. The left hand side of (1.26) is often used to dene the scattering
cross section, i.e.,
Z
0 () := tlim
jt(x)j2 dx :
(1.27)
!1
C
But in a scattering experiment it is certainly not the case that an experimenter measures at some large time t if the particle is located in some cone
C. However, comparing (1.27) with (1.14) one might expect that under
our assumptions on the behavior of the scattered particle, namely that it
1.4. THE FLUX-ACROSS-SURFACES THEOREM
25
moves eventually on a straight line, these two expression coincide by virtue
of Gauss' theorem and the fact that the particle leaves any bounded region of
space. In order to conrm this conjecture it turns out to be simpler to calculate the ux j t (x) directly using the splitting (1.24). Lets assume that the
second term in (1.24) does not only vanish in the L2 -sense but also doesn't
contribute to the ux, i.e., that
j t (x) = Im(t (x)rt(x)) xt t 3jb 0( xt )j2 :
Here we see that for large times the ux is directed outwards and j t is
normal to the surfaces R . Inserting this expression for the ux into the
right hand side of (1.14) a simple calculation and the substitution k := xt
yields the desired result:
Z1 Z
Z1 Z
Z1 Z
t
t
dt j d =
dt jj dj dt t 3jb 0( xt )j2 xt d
0
R
0
R
0
R
=
Z1
0
djkjjkj2
Z
jb 0(k)j2 d
=
Z
C
jb 0(k)j2 dk :
(1.28)
Herein d
denotes Lebesgue measure on the unit sphere 1. Thus we come to
the fundamental free ux-across-surfaces theorem that was rst proposed
by Combes, Newton and Shtokhamer in [9]: Let t = e iH0t 0, then for any
T 2 IR
lim
R!1
Z1 Z
Z1 Z
Z
t d j =
dt j t d = Rlim
dt
j
j
jb 0(k)j2 dk :
!1 T
T
R
R
C
(1.29)
The time T where the time integration starts should be thought of as the
time at which the detectors start counting. According to Remark 1.5 the
cross section should be independent of this choice, which is reected by the
above formula. This is because jb 0(k)j2 = jb t(k)j2 and thus the right hand
side of (1.29) is invariant under nite time shifts. It follows from (1.29) that
in the free case we arrive at a very simple expression for the scattering cross
section, namely at
Z
0
() = jb 0(k)j2 dk :
(1.30)
C
The free ux-across-surfaces theorem was rst proven by Daumer et al. in
[12] for initial wave functions 0 2 S , the space of Schwartz functions, and
later by Amrein and Zuleta in [5] for a dierent class of initial wave functions 0 . Both proofs essentially rely on the above calculation and the only
mathematical problem is to show that the remaining terms in (1.24) don't
26
CHAPTER 1. POTENTIAL SCATTERING
contribute to the ux through distant surfaces. In Section 2.2 we will present
a simplied version of the previous proofs which also shows that in (1.29)
the deviation from equality for nite R is essentially of order R 1.
Lets now turn to the general case with a short-range potential. From
what we learned in the last section it now seems obvious how to generalize
the free ux-across-surfaces theorem. Even if there is a localized potential
all states 0 2 Hcont (H ), i.e., the scattering states, will eventually, that is
for large times, evolve according to the free dynamics. Heuristically large
times correspond to large distances. Therefore the ux generated by e iHt 0
at large distances should be the same as that generated by e iH0 tW+ 10,
since both expressions are essentially equal, at least in the L2 -sense, for large
times. Hence we can now formulate the central theorem of this section,
the ux-across-surfaces theorem, also rst proposed by Combes, Newton
and Shtokhamer in [9]: Let t = e iHt 0, 0 2 Hac = RanW+ , then for all
T 2 IR
Z
Z1 Z
Z1 Z
t
t
dt jj dj = jW+d10 (k)j2 dk
lim
dt j d = Rlim
!1
R!1 T
C
T
R
R
and thus, with out := W+ 10,
0 () =
Z
C
jb out(k)j2 dk :
(1.31)
(1.32)
Remark 1.8. Recall again that according to Remark 1.5 the denition of
the scattering cross section (1.10) is independent of the choices for the origins
of space and time. To see that the expression on the right hand side of (1.32)
has the same symmetry note that, using the intertwining relation (1.19) and
the unitarity of W+, jW+d1t(k)j2 = jeik2=2W+d10 (k)j2 = jW+d10(k)j2 holds
for all t 2 IR. Also a spatial shift of the whole system along some y 2 IR3
only contributes with a factor eiky to the Fourier transform and thus doesn't
change its absolute value.
The ux-across-surfaces theorem was rst proven by Amrein and Zuleta in
[5] for a rather general class of short-range potentials. They manage to apply
standard methods from so-called time-dependent scattering theory in order
to establish (1.31) for a dense set of wave functions 0 . However, this set
contains only wave functions 0 such that b out is compactly supported away
from zero. From the physicist's point of view, it is no big deal to restrict
oneself, in the context of a Galilean invariant theory, to energies bounded
from above. Also the demand of excluding very small energies might seem
1.4. THE FLUX-ACROSS-SURFACES THEOREM
27
reasonable in a typical scattering situation. But in our heuristic discussion we
never made any such assumptions on the initial wave function and everything
we said until now (from the denition of the scattering cross section to its
expression in terms of (1.32)) could have been done analogously for a particle
starting at time 0 in the region where the potential is located and having also
arbitrarily small energies (which could correspond e.g. to a decay problem).
Since there are neither physical nor mathematical reasons why (1.31) should
only hold for this restricted class of wave functions, it is worthwhile to prove
it for a physically more reasonable class of wave functions. Therefore we
will establish (1.31) in Section 2.2 for potentials satisfying (V)4 and wave
functions 0 such that out 2 S , which again form a dense set in L2 (IR3 ).
Remark 1.9. Note that, in particular, this condition assures global exis-
tence of Bohmian mechanics whenever the potential satises the conditions
of Theorem 1.1. This is because all out 2 S are C 1-vectors of H0 and thus,
by virtue of the intertwining relations (1.20),
H n0 = H nW+ out = W+H0nout 2 L2
for all n 2 IN. Hence 0 2 C 1(H ).
One would clearly prefer a formulation with an explicit condition on 0
instead of demanding out 2 S . But the necessary mapping properties of
W+ have not been established yet for wave functions without energy cutos.
However, in Corollary 2.16 we will be able to nd sucient conditions on 0
assuring out 2 S by restricting ourselves to the smaller class of potentials
satisfying (V)1.
In the case of long-range potentials a related result has been established
by Amrein and Pearson [3]. As already mentioned, the wave operators do
not exist for Hamiltonians with a long-range potential. However, one can
still show the equality of the left hand sides of (1.26) and (1.31), i.e., that
Z
Z1
2
lim j (x)j dx = Rlim
t!1 C t
!1 T
dt
Z
R
j t d
Z1 Z
= Rlim
dt jj t dj :
!1 T
R
(1.33)
Again, the proof of (1.33) assumes compact support for b 0 not containing
the origin.
Remark 1.10.
The need for introducing so-called energy cutos, i.e.,
the demand that b out (k) is compactly supported away from the origin, goes
back to the fact that scattering theory was mainly concerned with proving
asymptotic completeness of the wave operators. There it usually suces to
prove certain propagation estimates for a dense subset of of wave functions
CHAPTER 1. POTENTIAL SCATTERING
28
in Hcont (H ) and to enlarge the domain by simple limiting procedures in L2
in the end. Assuming energy cutos makes it then possible to apply simple
but powerful integration by parts or \stationary phase" methods (see e.g. [22]
Theorem 3.2). However, in the case of (1.31) no such density arguments are
available, since the left hand side of (1.31) is unbounded as a sesquilinear
form of t for every nite R. For a more detailed discussion why removing
the energy-cutos is a nontrivial mathematical task see Section 2.2.
Let us shortly repeat what we established until now and why the uxacross-surfaces theorem is of central importance to scattering theory.
One reason is that the rst statement of (1.31) is equivalent to
R ) = lim IE0 (N R )
lim IE0 (Ntot
sgn
R!1
R!1
and thus implies the rst equality in (1.14). Hence the heuristic assumptions
leading to the expression (1.14) for the scattering cross section are justied
a posteriori. Since W+ 1 is unitary as a map from Ran(W+) to L2 (IR3)
Z
IR3
jb out (k)j2 dk = 1 :
Therefore the particle leaves any bounded region with probability 1 and the
scattering cross section dened by (1.10) is indeed a probability measure on
the unit sphere 1 .
At the same time the ux-across-surfaces theorem provides us also with
an explicit formula for calculating 0 , which, assuming quantum equilibrium
to hold, depends only on the wave function and not on the actual Bohmian
trajectories (as (1.10) does).
However, there is still one problem left. One has to connect (1.32) to
the usual quantum mechanical expressions for 0 . Therefore we will try to
simplify the expression on the right hand side of (1.32) by making further
assumptions which we expect to be valid in typical scattering experiments.
1.5 Initially free states and the S-matrix
In Section 1.3 we also introduced the scattering operator W = s limt! 1
eiHt e iH0t that connects a state 0 with its past asymptotic in := W 10.
In a typical scattering experiment the initial wave packet 0 is prepared far
away from the interaction region and therefore moves presumably according
to the free dynamics for some time. It also contains only particles that move
towards the scattering center and thus the free and the interacting evolutions
approximately coincide also for all earlier times. If these assumptions hold
1.5. INITIALLY FREE STATES AND THE S-MATRIX
29
true we can replace 0 in formula (1.31) by in = W 10 0, since the
L2 -norm of the dierence should become small:
kin 0k = k(W 1 1)0k = k(W 1)0k
= t!lim1 k eiHt e iH0 t 1 0 k
= t!lim1 ke iH0 t0 e iHt 0 k 0 :
(1.34)
This replacement leads to the central object of scattering theory, the so-called
scattering operator or S -matrix S := W+ 1W :
out = W+ 10 = W+ 1W in = S in :
(1.35)
Thus the formula for the scattering cross section now becomes
in () =
Z
C
jSdin(k)j2 dk :
(1.36)
This expression is, as it stands, only a reformulation of Equation (1.32). But
now we want to identify in with the initially prepared wave function 0 and
then (1.36) diers from (1.32) by an error bounded by k0 ink2 = k(W
1)ink2. In Section 2.3 we will show that this error becomes arbitrarily small
if one prepares in far away from the scattering region in a way that all
particles guided by in move essentially towards it. More precisely, we will
show that if b in(k) is supported in a half-space P := fk 2 IR3 : k 0g, an arbitrary unit vector, and nin(x) := in(x + n) is the translation of in
along the vector n, then
nlim
!1 k(W
1)nink ! 0 :
(1.37)
Thus in typical scattering situations where the initial wave packet is prepared
far away from the scattering center with all momenta pointing towards it, the
replacement 0 in is justied. However, one should keep in mind that
there are situations, as for example decay problems, where these assumptions
are not satised and one has to use (1.32) to calculate the cross section.
Remark 1.11.
For the proof of (1.37) we will apply standard methods
of time dependent scattering theory. Thus the class of allowed potentials is
larger than (V)2 and is in our case the class of Enss potentials (see Section
2.3 for details). The proof in Section 2.3 also nicely illustrates our point that
energy cutos are a useful tool whenever one proves estimates for bounded
operators in L2 : We will prove (1.37) rst for wave functions b in(k) compactly supported away from zero in a cone with opening angle smaller than .
30
CHAPTER 1. POTENTIAL SCATTERING
However, since (W 1)T n, where T n denotes the operator of translation
along n, is a uniformly bounded sequence of operators, we can conclude
that it approaches zero also on the closure of the above set.
Remark 1.12.
It is a standard result, used in the proof of asymptotic
completeness by the Enss method, that moving any 0 2 Hac backwards in
time according to the full dynamics leads to approximate eigenfunctions of
W , i.e., that
lim k(W 1)e iHt0 k = 0 :
(1.38)
t! 1
Heuristically, of course, large negative times correspond to large distances
from the scattering center. But in order to rigorously justify the identication of 0 with in we need the direct characterization in terms of spatial
and momentum support. However, from a mathematical point of view, the
problems are closely related. Hence, in Section 2.3, we will apply methods
similar to those used to prove (1.38).
Under the assumption of asymptotic completeness (actually it suces
to assume RanW = RanW+ ) the S -matrix, also called the S -operator or
scattering operator, is a unitary map form the Hilbert space underlying the
free dynamics to itself, which is in our case just L2 (IR3 ). It maps states
that are asymptotically free in the past into states that are asymptotically
free in the future and thus, in some sense, contains the information that is
accessible in a typical scattering experiment. It is a direct consequence of the
intertwining relations (1.19) that S commutes with the free time evolution
e iH0t and thus that in = in t , with in t = e iH0t in.
While in the case of Bohmian respectively quantum mechanics we were
able to derive and understand the S -matrix from a careful analysis of the
underlying dynamics, S is often, for example in quantum eld theories, the
only object accessible to calculation. This fact and the simplicity of formula
(1.36) together with the nice mathematical properties of S (e.g. unitarity) are
probably responsible for the central role S is playing in scattering theory. Our
derivation made clear that the formula (1.36) for the scattering cross section
is only valid under special assumptions, which are satised, however, in all
scattering experiments involving particles and interactions on the atomic
scale and macroscopic devices for preparation and detection.
1.6. SCATTERING OF ALMOST MOMENTUM EIGENSTATES
31
1.6 Scattering of almost momentum eigenstates and the connection to \naive" scattering theory
Until now we calculated the scattering cross section measure for an ensemble
of particles all prepared in a specic state in, i.e., all guided by the known
wave function in. Thus we naturally ended up with formulas explicitly
depending on the initial wave function in. But in many scattering experiments the exact form of the initial wave function is completely unknown.
What often is known very accurately is the momentum k0 of the particle,
i.e., one knows that b in(k) is mainly supported in a small neighborhood of
some xed k0. The form of in(x) is, at least on the scale dened by the
interaction, unknown and uncontrolled by the experimenter. However, this is
not a problem since it turns out that the experimental results depend mainly
on k0 but hardly on the precise form of in. This last fact is reected by
what we call the usual formulas of scattering theory.
We rst present shortly the main heuristic arguments leading to an expression for the scattering cross section independent of the shape of in and
then comment on the various ways to rigorously derive them from formula
(1.36).
The most naive but also most popular ansatz is to look for solutions of
the stationary Schrodinger equation
2
H (x; k0) = jk20j (x; k0)
that behave asymptotically for large jxj like
ijk0jjxj
(x; k0) eik0 x + f k0 (!) e
jxj ;
where x is written in spherical coordinates x = !jxj.8 The eik0x part represents the incoming plane wave with xed momentum k0 and f k0 (!) eijkjx0jjjxj
is a outgoing spherical wave with a direction dependent amplitude. By simply calculating the ux through some R R generated by the spherical
wave part, which is independent of R, one ends up with a formula for the
scattering cross section,
e k0 () =
Z
jf k0 (!)j2d
;
(1.39)
8 The choice of notation for will become clear in Chapter 2 when we deal with
generalized eigenfunctions.
32
CHAPTER 1. POTENTIAL SCATTERING
which only depends on k0. However, this expression is not normalized as
it stands and thus provides only relative probabilities. Since (x; k0 ) are
solutions of the time independent Schrodinger equation and since they are
supposed to describe the steady state of scattering of an endless beam of particles all with momentum k0, this approach is often called time-independent
scattering theory.
An expression equivalent to (1.39) can also be obtained by splitting S into
S =: 1 + T , where T denotes the so-called T -matrix, and formally evaluating
the T -part of (1.36) for in = eik0 x. This means that one ignores the contribution of the unscattered part of the wave and models the incoming state,
as in the time-independent approach, by a plane wave. However, since eik0x
is not in L2 , this procedure is merely formal: In momentum representation
S respectively T can be formally written
as integral operators with kernels
S (k; k0) = (k k0 ) 2i(jkj2=2 jkj02 =2)T (k; k0) (see e.g. [26], p. 108) and
a straight forward calculation then yields
e k0 () =
Z
j
T (!0jk0j; k0)j2d
0 :
(1.40)
Also this expression is not normalized, because we considered only the scattered part of in, given by T in.
Remark 1.13. The equivalence of (1.40) and (1.39) is due to the fact that
T (k; k0) = (2)
3
Z
e
iky V (y )
(y; k0) dy ;
(1.41)
and that is a solution of the Lippmann-Schwinger equation (see Chapter
2), from which
f k0 (!) = (2) 3
Z
e
ijk0 j!y V (y )
(y; k0) dy
(1.42)
follows.
There are various ways of deriving (1.40) and thus also (1.39) rigorously
from (1.36) that dier by the physical idealizations and assumptions involved,
but all end up with the same expression for . All results in the literature
start with rst averaging (1.36) over an ensemble of dierent initial wave
functions or an ensemble of dierent scattering centers. In [2] it is shown, for
example, that if b in(k) is supported in a neighborhood of some k0, one can
rst average the T -part of (1.36) over an ensemble ain, a 2 IR2, where ain
denotes the translation of in along a vector a in the plane perpendicular to
k0, and then take the limit jb in(k)j2 ! (k k0) in order to obtain (1.40).
The ensemble is interpreted as an incoherent beam of particles.
1.6. SCATTERING OF ALMOST MOMENTUM EIGENSTATES
33
We will show in Section 2.5, however, that the averaging procedure is
not necessary, i.e., that one can get (1.40) from (1.36) directly by taking
the limit jb in(k)j2 ! (k k0). More precisely we will show that for certain
sequences n of wave functions with limn!1 jb n(k)j2 = (k k0) in the sense
of distributions the normalized T -part of (1.36) converges to the normalized
version of (1.40):
k0 () =
R jTd
n(k)j2 dk R jT (!0jk0j; k0)j2 d
0
C
=R
:
nlim
!1 R 3 jTd
n(k)j2 dk 1 jT (!0jk0j; k0)j2 d
0
IR
(1.43)
It is necessary to normalize the T -part of (1.36) before taking the limit because the probability for the particle to be scattered at all, that is the norm
of T n, approaches zero when n gets spread out wider and wider in space
with growing n. Thus it is important to note that the formula (1.40) only
gives the correct relative probabilities for solid angles not containing the
direction k0=jk0j of the incoming particle. In real scattering experiments this
is reected by the fact that only few particles out of a beam are scattered
and the detectors are placed away from the original direction of the beam.
Remark 1.14.
The validity of (1.36) was based on the assumption
that k(W 1)ink 0. In order to derive (1.40) directly from (1.32)
one therefore has to look at a sequence n which satises in addition to
limn!1 jb n(k)j2 = (k k0) also limn!1 k(W 1)nk = 0. Since n(x) is
spread out wider and wider in space with growing n, it is not entirely obvious
that one can do both limits consistently at the same time. We will show in
Section 2.5, however, that this is actually no problem. Practically speaking
(1.40) is a good approximation if jb in(k)j2 is well localized around k0 but
jin(x)j2 is still mainly supported far away from the scattering center.
With the derivation of formula (1.43) we provided the rst completely
rigorous derivation and thus justication of the commonly used formulas
(1.39) and (1.40) for the scattering cross section. While most of the single
steps we made in this chapter were proven in some way or another before, the
overall coherent picture is certainly new and worth the mathematical eorts
of the following chapter.
34
CHAPTER 1. POTENTIAL SCATTERING
Chapter 2
Mathematical results
2.1 Regularity of the generalized eigenfunctions
In this section we introduce the so-called generalized eigenfunctions (x; k).
They will be applied throughout this chapter in various ways. For some
applications we need certain regularity properties of (x; k) that have not
been established yet and will be summarized in Theorem 2.4. We will begin
with a short introduction and a proposition summing up important known
properties of the generalized eigenfunctions.
The free Hamiltonian is diagonalized by the Fourier transformation, i.e.,
by an expansion in terms of plane waves eikx. A Hamiltonian of the form
H = H0 + V , where V is a scattering potential, can be diagonalized by an
expansion in terms of the generalized eigenfunctions (x; k). The plane
waves are solutions of the time-independent free Schrodinger equation
1 eikx = jkj2 eikx ;
2
2
where we wrote 12 instead of H0, since H0 denotes the Hamiltonian as an
operator on L2 and the functions eikx are not members of this Hilbert space.
Since in the short-range case the potential V (x) is localized, one expects that
there are solutions (x; k) of the time-independent Schrodinger equation
with potential V ,
2
( 12 + V (x)) (x; k) = jk2j (x; k) ;
that still behave asymptotically like plane waves in an appropriate sense.
One naturally expects that they approach plane waves for large jxj as well
35
36
CHAPTER 2. MATHEMATICAL RESULTS
as for large jkj, at least if the potential is bounded. We will see that these
locally distorted plane waves (x; k) are as helpful in the analysis of the
time evolution e iHt as the plane waves are in the case of e iH0t .
In order to make the above statements precise one turns Schrodinger's
equation into an integral equation, the so-called Lippmann-Schwinger equation. Formally (because are not expected to be members of L2 ) we write
2
(H0 + V ) = jk2j and thus
2
(H0 jk2j ) = V :
Now we think of the term on the right hand side as an inhomogeneity and
solve the equation by inverting (H0 jkj2=2) and taking care of the boundary
conditions by adding eikx, which is a solution of the homogeneous equation:
2
= eikx (H0 jkj ) 1V :
(2.1)
2
The resolvent (H0 jkj2=2) 1 doesn't exist as a bounded operator on L2 since
the spectrum of H0 is [0; 1). But one can make sense of the above equation
in various ways. In three space dimensions the kernel of (H0 + 2=2) 1 can
be computed explicitly for 2 C,
l Re > 0, i.e., 2 =2 in the resolvent set of
H0:
Z jx yj
2
(H0 + 2 ) 1(x) = (2) 1 ejx yj (y) dy :
Inserting this expression into Equation (2.1), formally taking = ijkj,
one arrives at the Lippmann-Schwinger equations. The following proposition
states that the equation has solutions with exactly those properties one would
expect from our purely formal derivation.
Proposition 2.1. Let V satisfy (V)2. Then
for any k 2 IR3 nf0g there are
unique continuous solutions (; k) : IR3 ! Cl of the Lippmann-Schwinger
equations
Z ijkjjx yj
(x; k) = eikx 1 e
V (y)(y; k) dy;
(2.2)
2 jx yj
with the boundary conditions limjxj!1[ (x; k) eikx] = 0, which are also
classical solutions of the stationary Schrodinger equation
1
2
+ V (x) (x; k) = jkj (x; k);
(2.3)
2
2
such that:
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 37
(i) For any compact D IR3 nf0g the functions (; ) : IR3 D ! Cl are
uniformly continuous.
(ii) For any f 2 L2 (IR3) the generalized Fourier transforms
Z
1
(Ff )(k) =
l:i:m: (x; k)f (x) dx
(2) 23
exist in L2(IR3)1 .
(iii) RanF = L2 (IR3) and F : Hac(H ) ! L2 (IR3) are unitary and the
inverse of F is given by
Z
1
1
(F f )(x) =
l:i:m: (x; k)f (k) dk :
(2) 32
(iv) For any f 2 D(H ) \ Hac (H ) we have
Z k2
1
Hf (x) = (F 2 Ff )(x) =
l:i:m: 2 (x; k)(Ff )(k) dk
(2) 32
(2.4)
and therefore for any f 2 Hac(H )
1k
e
iHt f (x)
2
2
= (F 1e i k2 tFf )(x)
Z k2 t
1
=
l:i:m: e i 2 (x; k)(Ff )(k) dk:
(2) 32
(v) For any f 2 Hac(H ) the relations Wf = F 1F f hold, where F
denotes the ordinary Fourier transformation.
The proof of Proposition 2.1 is due to Ikebe and can be found in [18].
Remark 2.2. Similar eigenfunction expansions can be obtained also for
potentials with slower decay. But then, in general, the continuity in k will
not hold any more (see, e.g., [1]).
Remark 2.3. In order to obtain similar expansions in space dimensions
dierent from three or for potentials with slower decay, one can't use the
1 l:i:m: R is a short way of writing s limR!1 R , where s lim denotes the limit in
BR
L2 .
38
CHAPTER 2. MATHEMATICAL RESULTS
explicit form (2.2) of the Lippmann-Schwinger equation anymore. Instead
the operator (H0 jkj2=2) 1 is dened as a limit Re ! 0 of (H0 + 2 =2) 1
on suitably enlarged spaces, usually weighted L2 spaces.
In order to prove the ux-across-surfaces theorem we will use the generalized Fourier representation of the time evolved wave function e iHt 0(x)
(see Proposition 2.1.(iv)). But in contrast to the situation of the free time
evolution, the generalized eigenfunctions are not known explicitly, but only
implicitly as the solutions of the Lippmann-Schwinger equation. Thus, to
make use of the generalized Fourier representation, we need to know more
about the regularity of (x; k).
Since the functions (; k) are solutions of the time independent Schrodinger equation, the regularity of (x; k) as functions of x with xed k =
6
0 depends only on the regularity of V (x). Standard methods for elliptic
partial dierential equations are available in order to make this connection
precise. However, for us the continuity in x as expressed in Proposition 1.(i)
is sucient. Our concern will be the question of the regularity of (x; k)
as functions of k, which turns out to be mathematically much more subtle.
By regularity we actually mean several things. In order to do integration
by parts in expressions like the right hand side of Proposition 1.(iv) one
needs dierentiability of (x; k) with respect to k. However, because of the
appearance of jkj in the Lippmann-Schwinger equation (2.2) dierentiability
of (x; k) with respect to k can only be expected for k 6= 0. It also turns
out to be very helpful in calculations to get rid of the l:i:m: in the denition
of the generalized Fourier transformation on L2 and to dene it, as in the
case of the ordinary Fourier transformation, directly on L1 . Therefore we
need to show that (x; ) 2 L1. Last but not least we also have to control
integrals over derivatives of (x; k) with respect to k and thus need also
upper bounds on these derivatives.
Before we state the main theorem of this section we have to introduce an
additional condition on V that we will need in order to prove that are
bounded.
Mathematically, zero is said to be a resonance of H if there exists a
solution f of 12 f (x) + V (x)f (x) = 0 such that hxi f (x) 2 L2 (IR3) for
any > 21 and not for = 0 (see [20]). Here and in the following we
abbreviate hxi := 1 + jxj.
Heuristically one can picture the situation as follows: when continuously
changing only the strength of a potential that allows for bound states with
negative energies, but not its shape, the eigenvalues also change continuously.
If a negative eigenvalue moves towards zero three things can happen: One
might eventually be left with a potential having a bound state with zero
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 39
energy, i.e., a Hamiltonian with eigenvalue zero, or a resonance at zero might
appear or neither of both. In the case of a zero energy eigenvalue or resonance
the spectral density at zero diverges (see [20]). Since the spectral density
is closely related to the generalized eigenfunctions (see e.g. [26], Theorem
XI.41.(d)), they presumably also become unbounded at k = 0.
This heuristic picture already indicates that the appearance of zero-energy
resonances or eigenvalues is an exceptional event: Hc = H0 + cV can have
a zero-energy resonance or eigenvalue only for c in discrete subset of IR (see
[20]).
Now we are ready to state the main theorem of this section:
Theorem 2.4. Let the potential satisfy the condition (V)n for some n 3,
n 2 IN. Then
(i) (x; ) 2 C n 2(IR3 n f0g) for all x 2 IR3 and the partial derivatives
@k (x; k), jj n 2, are continuous with respect to x and k.2
(ii)
lim sup hxi jjj@k (x; k) @k eikxj = 0
jkj!1 x2IR3
for every jj n 2.
(iii) If, in addition, zero is not an eigenvalue or a resonance of H , then
(x; k) are continuous on IR3 IR3 and
sup
x2IR3 ;k2IR3
j(x; k)j < 1
and for any with jj n 2 there is a c < 1 such that
sup
j@k (x; k)j < chxijj :
3
k2IR nf0g
Remark 2.5.
In particular, statement (ii) with jj = 0 conrms the
heuristic insight that the generalized eigenfunctions approach plane waves
also for large jkj uniformly in x. Also the bounds in part (iii) are exactly
those which hold for plane waves.
2 We use the usual multi-index notation: = (1 ; 2 ; 3 ), i 2 IN0 , @ f (k) :=
k
1
@k1 @k22 @k33 f (k) and jj := 1 + 2 + 3 .
40
CHAPTER 2. MATHEMATICAL RESULTS
Theorem 2.4 provides the basis for the proof of the ux-across-surfaces
theorem in Section 2.2, which was for us the original motivation for looking
at the regularity of the generalized eigenfunctions in k. However, it turns out
that this result is of considerable interest in its own. It immediately implies
two interesting corollaries. According to Remark 1.13 the T -matrix can be
expressed in terms of the generalized eigenfunctions and thus the regularity
of (x; k) implies some regularity for T . This corollary will be stated in
Section 2.4, where it is applied in the proof of the main result of that section.
The second corollary states that the Riemann-Lebesgue lemma holds also
for the generalized Fourier transformation. Furthermore, the dierentiability
of the generalized Fourier transform of a function is connected to its decay
as in the case of the ordinary Fourier transform. We will apply this result in
the proof of Corollary 2.16. Related results can be found in Isozaki [19].
Corollary 2.6. Let V satisfy (V)n with some n 3 and let zero not be
an eigenvalue or resonance of H . Then, for any N n 2 and any f such
that hxiN f 2 L1 (IR3), we have that Ff 2 C N (IR3) and @k Ff 2 C1(IR3 )
for all with jj N .3
Proof. Let hxiN f 2 L1 , 0 jj N , then
Z
Z
@k (F+f )(k) = @k 1 3 +(x; k)f (x) dx = 1 3 @k + (x; k)f (x) dx
(2) 2
(2) 2
Z
Z
=: 1 3 @k e ikxf (x) dx + 1 3 @k + (x; k)f (x) dx(2.5)
(2) 2
(2) 2
is bounded and continuous since j@k +(x; k)j is bounded by chxijj according to Theorem 2.4.(iii) and hxijjf 2 L1 by assumption. Furthermore, the
rst term in the second line belongs to C1 by the ordinary Riemann-Lebesgue
lemma and the second term belongs to C1 since hxi jjj@k + (x; k)j tends
uniformly to zero for jkj ! 1 according to Theorem 2.4.(ii).
Proof [of Theorem 2.4]. To simplify notation we will give the proof for
+(x; k) =: (x; k) since the proof for (x; k) is exactly the same apart
from the change of some signs.
The structure of the proof will be as follows: First we introduce some
notation and results from Ikebe and Povzner that we will use frequently.
Then part (i) of Theorem 2.4 is shown for jj = 1 involving several lemmas
3 Recall that C1 denotes the set of continuous functions vanishing at innity.
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 41
and results proven in the appendix. The generalization to jj 1 will be
sketched afterwards.
The proof of part (ii) will involve an expansion of (x; k) for large jkj
related to the Born series. In order to conclude statement (iii) we will be left
with an analysis of (x; k) near jkj = 0.
We start with an investigation of Equation (2.2). If (x; k) = eikx +
(x; k) is a continuous solution of the Lippmann-Schwinger equation (2.2)
with limjxj!1 (x; k) = 0 for k 2 IR3 , then (x; k) is a solution of the integral
equation
Z e ijkjjx yj
h
i
1
(x; k) = 2 jx yj V (y) eiky + (y; k) dy
(2.6)
and (; k) 2 C1(IR3 ). Therefore Equation (2.6) is examined on the Banach
space B = C1(IR3), the set of continuous functions vanishing at innity,
equipped with the norm kf kB = supx2IR3 jf (x)j. L(B ) denotes the space of
bounded linear operators mapping B into itself, equipped with the operator
norm. Following Ikebe [18] we dene the linear operators Tk 2 L(B ), k 2 IR3 ,
by
1 Z e ijkjjx yj V (y)f (y) dy :
(2.7)
2 jx yj
Since we will make use of some results of Ikebe and Povzner, we state them
as a lemma:
(Tk f )(x) =
Lemma 2.7. Let the potential satisfy the condition (V)3. Then:
(i) The operator Tk 2 L(B ) dened in (2.7) is compact for all k 2 IR3.
(ii) Let f (x) be a bounded continuous function on IR3, then
Z ijkjjx yj
h(x) := 21 ejx yj V (y)f (y) dy
is an element of B for all k 2 IR3 and h(x) = O(jxj 1) for jxj ! 1.
(iii) Let
Z ijkjjx yj
g(x; k) := 21 ejx yj V (y)eiky dy = Tk eik (x) ;
then g(; k) 2 B for all k 2 IR3 and g(; k) is continuous with respect to
k.
42
CHAPTER 2. MATHEMATICAL RESULTS
(iv) Let f (; k) 2 B be a solution of the homogeneous equation f (; k) =
Tk f (; k) for k 2 IR3 . If jkj > 0 then f = 0 and if k = 0 then ( 12 +
V (x))f (x; 0) = 0.
(v) The map T : IR3 ! L(B ), k 7! Tk is continuous.
For the proofs of (i), (ii), (iii) and (iv) see Ikebe [18], for the proof of (v) see
Povzner [23].
Since we will use similar reasoning, we will briey repeat Ikebe's proof of
the existence of continuous solutions of Equation (2.6) starting from Lemma
2.7. In the notation just introduced, Equation (2.6) now reads
(; k) = g(; k) + Tk (; k) :
(2.8)
According to Lemma 2.7.(iv) the homogeneous equation (; k) = Tk (; k)
has only the trivial solution (x; k) = 0 if k 6= 0. Thus 1 is not an eigenvalue
of Tk and therefore in the resolvent set since Tk is compact (see e.g. [30]),
i.e., (1 Tk ) 1 2 L(B ) exists. The unique solution of (2.6) for jkj > 0 is then
given by
(; k) = (1 Tk ) 1g(; k) :
(2.9)
Since L(B ) is a Banach algebra in which the map A 7! A 1 is continuous
(see e.g. [30]), from Lemma 2.7.(v) it follows that (1 Tk ) 1 is continuous in
k. Thus, since according to Lemma 2.7.(iii) g(; k) is continuous with respect
to k, we have that (x; k) is continuous with respect to k.
We will now prove part (i) of Theorem 2.4 for jj = 1 and assume (V)3.
The generalization to jj > 1 will then be immediate. Consider arbitrary
l 2 f1; 2; 3g and k0 2 IR3 n f0g. We will show that @kl (x; k) exists and is
continuous for k in some neighborhood of k0 and x 2 IR3 .
We use the following notation: kl denotes the l-th Cartesian coordinate
of a vector k 2 IR3 and kl the tuple of the other coordinates. Symbolically
we will write k = (kl ; kl).
By (formally) dierentiating (2.6) we obtain
@ (x; k) = @ g(x; k) + i kl Z e ijkjjx yjV (y)(y; k) dy
@kl
@kl
2 jkj
Z
i
j
k
jj
x
yj
1 e
@ (y; k) dy :
V
(
y
)
(2.10)
2 jx yj
@k
l
Assume that for kl 2 Il := [kl0 l ; kl0 + l] and kl 2 Il := [kl0 l ; kl0 + l] where
l and l are chosen such that in particular 0 2= I := Il Il , the equation
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 43
(x; k) = @k@ g(x; k)
l
2
3
kl
Z
Z
6
7
+ 2i jkklj e ijkjjx yjV (y) 64 (y; (kl0; kl)) dkl0 + (y; (kl0; kl))75 dy
kl0
1 Z e ijkjjx yj V (y) (y; k) dy ;
(2.11)
2 jx yj
R
which arises from (2.10) by substituting for (x; k) = kkl0l (x; (kl0; kl)) dkl0+
(x; (kl0; kl )), has a continuous solution (x; k). Integrating (2.11) with respect to kl and using Fubini's theorem we get
Zkl
kl0
(x; (kl0; kl)) dkl0 = g(x; k) g(x; (kl0; kl))
2
0 k0
13kl
l
Z
Z
1 V (y) 66 e ijkjjx yj B
00; k )) dk00 + (y; (k0; k ))C
B
CA775 dy
(
y;
(
k
l
l
l
4
@
l
l
2
jx y j k 0
= g(x; k)
kl0
l
g(x; (kl0; kl))
0
1
kl
Z
Z
i
j
k
jj
x
y
j
B
1 e
0; k )) dk0 + (y; (k0; k ))C
B
V
(
y
)
(
y;
(
k
l
l
l l C
@
l
A dy
2 jx yj
0
kl
0l ;k )jjx yj
Z
i
j
(
k
l
(2.12)
+1 e
V
(y)(y; (kl0; kl )) dy :
2
jx y j
Since (x; (kl0; kl )) is a solution of (2.6) the second and fourth term of the
right hand side of (2.12) combine to (x; (kl0 ; kl)). Therefore (2.12) simply
reads
Z
kl
(x; (kl0; kl )) + 0 (x; (kl0; kl )) dkl0
kl
1 Z e ijkjjx yj
=
Z
kl
(y; (kl0; kl)) + 0 (y; (kl0; kl )) dkl0
kl
!
g(x; k) 2 jx yj V (y)
dy :
In other words, if (x; k) is a Rcontinuous solution of (2.11) then the function f (x; k) = (x; (kl0; kl )) + kkl0l (x; (kl0; kl)) dkl0 is a solution of Equation
CHAPTER 2. MATHEMATICAL RESULTS
44
(2.6). Since (2.6) has the unique solution (; k) in B we may conclude that
f (x; k) = (x; k), i.e., that @kl (x; k) = (x; k) for x 2 IR3 and k 2 I once we
have shown that f (; k) 2 B .
We show now that equation (2.11) has a solution (x; k) which is continuous with respect to x 2 IR3 and k 2 I such that
Z
kl
(; (kl0; kl )) + 0 (; (kl0; kl)) dkl0
kl
2B:
From the physical argument that (x; k) eikx + eijjkxjjjxj for jxj ! 1 we
expect jrk (x; k)j eijkjjxj, jxj ! 1, to be a uniformly bounded function,
but we will only show that jrk (x; k)j c(1 + jxj)s for any s > 0. Hence we
start by multiplying equation (2.11) by hxi s := (1 + jxj) s, s > 0:
Z ijkjjx yj
e(x; k) = hxi s@kl g(x; k) 21 hexisjx yj hyisV (y)e(y; k) dy
2
3
Z e ijkjjx yj
66Zkl e 0
i
k
l
s
0 + hy i s (y; (k0; k ))77 dy :
+ 2 jkj
h
y
i
V
(
y
)
))
dk
(
y;
(
k
;
k
l
l l 5
l
4
l
hxis
0
k
l
where we abbreviated e(x; k) := hxi s (x; k). To see that
(x; k) = hxise(x; k) c(1 + jxj)s ;
we show that (2.13) has a unique solution in
sup jf (x; k)j = 0g
Be := ff (x; k) 2 C (IR3 I ) : jxlim
j!1
(2.13)
k2I
In Section 2.5 we prove that Be equipped with the norm
kf kBe = sup
jf (x; k)j
3
x2IR ;k2I
is a Banach space (see Lemma 2.27 in Section 2.5) and that for f 2 Be the
operators
1 Z e ijkjjx yj hyisV (y)f (y; k) dy ;
2 hxisjx yj
Z ijkjjx yj
(Te0f )(x; k) := i kl e s hyisV (y)f (y; k) dy ; and
2 jkj
hxi
Z kl
0
0
f )(x; k) :=
(Kf
0 f (x; (kl ; kl )) dkl ;
e )(x; k) :=
(Tf
kl
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 45
belong to L(Be ) if s > 0 is chosen such that hxisV (x) still satises (V)3
(Lemma 2.28 in Section 2.5).
Noting that hxi s@kl g(x; k) and hxi s(x; (kl0; kl )) belong to Be (Lemma
2.28), equation (2.13) can be written as
fe + Te0hi s(; (kl0; kl)) + Te e :
e = hi s@kl g + Te 0K
where hi s denotes the operator of multiplication with hxi s in Be . To prove
that this equation has a unique solution in Be we show that (1 Te) 1 2 L(Be )
exists and that
fe (2.14)
e = (1 Te) 1 hi s@kl g + Te0hi s(; (kl0; kl )) + (1 Te ) 1Te 0K
has a unique solution. The former will be content of Lemma 2.8, and to see
the latter note that according to Lemma 2.28.(i)
k(1 Te ) 1Te 0KfkL(Be) k(1 Te) 1Te0kL(Be) 2l :
Also k(1 Te ) 1kL(Be) and kTe 0kL(Be) depend on l since the space Be itself
depends on l . But the norm of these operators decreases as l decreases since
according to Lemma 2.27.(ii) and the constructions
in the proofs of Lemma
2.28 and Lemma 2.8 kTe 0kL(Be) supk2I kTk0skL(B) and k(1 Te ) 1kL(Be) supk2I k(1 Tks) 1kL(B) . Thus one can choose l such that
k(1 Te) 1 Te 0KfkL(Be) < 1 :
f is a contraction
Then (2.14) has a unique solution e 2 Be since (1 Te) 1 Te 0K
in a complete metric space.
Now (x; k) = hxise(x; k) is a solution of (2.11) and f (x; k) = (x; (kl0 ; kl))
R
+ kkl0l (x; (kl0; kl ))dkl0 is a solution of (2.6). Recall that to conclude f (x; k) =
(x; k), i.e., that is the partial derivative of with respect to kl , we
need to show f (; k) 2 B . By construction supk2I j (x; k)j = O(jxjs) for
jxj ! 1 and therefore also jf (x; k)j = O(jxjs) for any k 2 I . Thus writing
V (x)f (x; k) = hxisV (x) hxi sf (x; k) and observing that hxisV (x) satises
(V)3 and hxi sf (x; k) is bounded we use Lemma 2.7.(ii) to conclude that
f (; k) 2 B .
To complete the proof of part (i) for jj = 1 we need to show the following
lemma:
Lemma 2.8. (1 Te) 1 2 L(Be ) exists.
CHAPTER 2. MATHEMATICAL RESULTS
46
Proof. First we show that (1 Tks) 1 2 L(B ) exists, where Tks :=
hi sTk his 2 L(B ). Since hxisV (x) meets the requirements of Lemma 2.7
and multiplication by hxi s is a bounded operation on B , Tks is compact.
Therefore (1 Tks) 1 exists if the homogeneous equation (1 Tks)fs = 0
has only the trivial solution fs = 0. Now let fs 2 B be a solution of the
homogeneous equation which explicitly reads
Z e ijkjjx yj
1
s
fs(x) = (Tk fs)(x) = 2 hxisjx yj V (y)hyisfs(y) dy :
Then f (x) := hxisfs(x) is a solution of
Z e ijkjjx yj
1
f (x) = 2 jx yj V (y)f (y) dy
and Lemma 2.7.(ii) implies f 2 B since hxisV (x) satises (V)3 and hxi sf (x)
is bounded. Using Lemma 2.7.(iv) we conclude that f (x) = fs(x) = 0 for
k 6= 0. Therefore (1 Tks) 1 exists for any k 2 I . The continuity of (1 Tks) 1
with respect to k follows again from the continuity of (1 Tks).
Using Lemma 2.27.(ii) we dene the operator (1 Te ) 1 2 L(Be ). Since
(1 Tks)(1 Tks) 1 = (1 Tks) 1(1 Tks) = 1 holds for all k 2 I , also
(1 Te)(1 Te) 1 = (1 Te ) 1(1 Te) = 1 holds.
It is now easy to prove the existence of higher order derivatives by induction. From the proof for jj = 1 we conclude that if (x; k) 2 B is a solution
of (2.6) then hxi s@kl (x; k) is given by the unique solution (x; k) in B of
Z ijkjjx yj
(x; k) = hxi s@kl g(x; k) + 2i jkklj e hxis V (y)(y; k) dy
1 Z e ijkjjx yj hyisV (y) (y; k) dy
(2.15)
2 hxisjx yj
for any k 2 IR3 n f0g.4 In general, assume that (x; ) 2 C p(IR3 n f0g) for
some p < n 2 and that hxi s p+1@k (x; k), jj = p, is given by the unique
solution (x; k) of
(x; k) = hxi
s p+1 [@ g (x; k) + @ (T )(x; k)
k
k k
+(Tks+p 1 )(x; k)
(Tk @k )(x; k)]
(2.16)
4 To see that (2.15) has a unique solution in B recall that we established in the proof
of Lemma 2.8 also the existence of (1 T s ) 1 .
k
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 47
in B , where Tks+p 1 is given by
1 Z e ijkjjx yj V (y)hyis+p 1f (y) dy :
2 hxis+p 1jx yj
(Tks+p 1f )(x) :=
Then one can prove by exactly the same method as in the case jj = 1 that
@kl (x; k) exists: Equation (2.16) is analogous to (2.8) where g is replaced by
hxi s p+1 [@k g(x; k) + @k (Tk )(x; k) (Tk @k )(x; k)] 2 B and Tk by Tks+p 1.
As long as the modied potential hyis+p 1V (y) satises (V)3 the proof of
dierentiability of the solution of (2.16) can be done along the same lines as
for jj = 1.
Proof [of part (ii) of Theorem 2.4].
function (x; k) = (x; k)
Recall (2.9):
eikx
First we will show that the
converges uniformly to zero for jkj ! 1.
(; k) = (1 Tk ) 1 g(; k) :
We shall show that limjkj!1 kg(; k)kB = 0, but we have no simple control of
the norm of (1 Tk ) 1, for example in terms of the Born series, since
Z
kTk kL(B) = sup3 jjxV (yy)jj dy = const.
x2IR
does not depend on jkj. Following Zemach and Klein [31] we iterate Equation
(2.8) once and obtain
(; k) = g(; k) + Tk g(; k) + Tk2(; k) ;
with the formal solution
(2.17)
(; k) = (1 Tk2) 1(g(; k) + Tk g(; k)) :
(2.18)
If equation (2.17) has a unique solution it must equal the unique solution of
(2.8) since any solution of (2.8) is clearly also a solution of (2.17). We shall
now rst establish that a) (1 Tk2) 1 ! 1 for jkj ! 1 and then that b)
kg(; k) + Tk g(; k)kB ! 0 for jkj ! 1, since then
lim k(; k)kB jklim
k(1 Tk2) 1kL(B) k(g(; k) + Tk g(; k)kB = 0 :
j!1
jkj!1
a) follows from
Lemma 2.9. Let V 2 L1 \ L2 then
CHAPTER 2. MATHEMATICAL RESULTS
48
lim kTk2kL(B) = 0 :
jkj!1
(2.19)
(2.19) also holds, if Tk2 is understood as an operator on bounded continuous
functions.
Now jkj can be chosen such that kTk2kL(B) < 1 and then (1 Tk2) 1 is
given as the norm convergent Born series:
(1 Tk2) 1 =
Thus limjkj!1 k(1 Tk2)
1
1
X
(Tk2)n :
n=0
1kL(B) = 0.
Proof [of Lemma 2.9]. We compute for f 2 B
1 Z
42
Z
= 412
Z
1
= 42
Z
=: 412
(Tk2 f )(x) =
Z ijkjjy zj
V
(y) e
jx yj
jy zj V (z)f (z) dz dy
Z ijkj(jx yj+jy zj)
V (z)f (z) ejx yjjy zj V (y) dy dz
"
#
V (z) f (z) jx zj Z e ijkj(jx yj+jy zj) V (y) dy dz
jx z j
jx yjjy zj
V (z) f (z)I V (x; z; jkj) dz :
jx z j
e
ijkjjx yj
Zemach and Klein [31] showed that for V 2 C01 (IR3 )
lim sup jI V (x; z; jkj)j = 0 ;
jkj!1 x;z2IR3
i.e., that limjkj!1 kTk2kL(B) = 0 holds for V 2 C01 (IR3 ).
Potentials V 2 L1 \ L2 will be approximated in the following norm:
Z
kjV kj = sup3 jjxV (yy)jj dy :
x2IR
Observing that
Z
Z V (y)
V
(
y
)
kjV kj sup3
dy + sup3
dy
x2IR jx yj<1 jx y j
x2IR jx yj1 jx y j
c (kV kL2 + kV kL1 ) < 1 ;
(2.20)
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 49
where we used Schwarz's inequality for the kV kL2 term, we conclude that,
since any V 2 L1 \ L2 can be approximated by a function U 2 C01 simultaneously in L1 and L2 -norm, this is also true for the kj kj-norm. Thus we get
the following bound for the norm of Tk2:
Z e ijkjjx yj
1
2
kTk f kB = sup3 (2)2 jx yj (V (y) U (y) + U (y))
x2IR
Z e ijkjjy zj
jy zj V (z)f (z) dz dy
!
U
kf kB kjV kj kjV U kj + sup 3 jI (x; z; jkj)j :
x;z2IR
The rst term in the brackets becomes small for appropriately chosen U 2 C01
while the second one converges to zero for jkj ! 1.
We now proceed to b) namely that limjkj!1 kg(; k) + Tk g(; k)kB = 0.
By Tk g(x; k) = Tk2eikx and Lemma 2.9 limjkj!1 kTk g(; k)kB = 0 follows
immediately. To get the analogous statement for g(; k) we assume rst again
V 2 C01 (IR3). Then
Z e ijkjjx yj
1
jg(x; k)j = 2 jx yj V (y)e ik(x y) dy
Z
i
j
k
jj
z
j
(1+cos
)
21 e jzj
V (x z) dz
<0
Z jV (x z)j
+ 21
dz
j
z
j
0
(2.21)
holds. Herein denotes the angle between z = (x y) and k. Stationary
phase methods on the rst term yields
!
Z
1 1
d e ijkjjzj(1+cos ) V (x z)jzj djzjd
(z)
2 <0 ijkj(1 + cos ) djzj
!
Z
= 21 ijkj(1 +1 cos ) e ijkjjzj(1+cos ) djdzj V (x z)jzj djzjd
(z)
<0
50
CHAPTER 2. MATHEMATICAL RESULTS
Z d
1
1
2 jkj(1 + cos ) djzj V (x z)jzj djzjd
(z)
0
! 0;
jkj(1 +ccos ) jkj!1
0
where
Z djdzj V (x z)jzj dz = c < 1
sup jzj2
x2IR3 was used. This follows directly from V 2 C01.
The second term in (2.21) is an integration over a cone with opening angle
0 where the potential in the integrand has compact support, is bounded and
displaced by x. Thus
Z jV (x z)j
Z
dz
c
lim
j sin j d = 0
lim
sup
0 ! 0
0 ! x2IR3 0
jzj
and limjkj!1 supx2IR3 jg(x; k)j = 0 follows. To get this for V 2 L1 \ L2 we
approximate V by U 2 C01 as in Lemma 2.9.
Analogously we show that hxi jjj@k (x; k) @k eikxj vanish uniformly for
jkj ! 1. According to the proof of part (i) of Theorem 2.4 @k (x; k) is
obtained as the unique solution of (2.16) in B . For large jkj the operator
(1 (Tks+p 1)2 ) 1 with p = jj can be expanded in terms of the Born series
and the solution of the modied equation is given by
= (1 (Tks+p 1)2) 1 hi s p+1 [@k g + @k Tk Tk @k ]
+ Tks+p 1hi s p+1 [@k g + @k Tk Tk @k ] :
It can be shown by the same methods as in the case of g(x; k) that the term,
on which (1 (Tks+p 1)2) 1 is acting, uniformly approaches zero for jkj ! 1.
This completes the proof of part (ii).
Proof [of part (iii) of Theorem 2.4]. From the continuity of and the
fact that limjxj!1 ((x; k) eikx) = 0 for all k 6= 0 Ikebe already concluded
that for compact D IR3 n f0g
sup j(x; k)j < 1
x2IR3 ;k2D
holds. Since the bounds on and its derivatives for jkj large enough are
already covered by part (ii), it remains to examine the case jkj ! 0.
2.1. REGULARITY OF THE GENERALIZED EIGENFUNCTIONS 51
If H has a zero-energy resonance or eigenvalue, according to Kato [20], the
spectral density is singular at E = 0. Since the spectral density and the generalized eigenfunctions are closely related (see e.g. [26], Theorem XI.41.(d)),
we expect that in this case also the generalized eigenfunctions become singular at k = 0.
But assuming that H has neither a resonance nor an eigenvalue at E = 0,
the eigenfunctions stay bounded near k = 0:
Proposition 2.10. Let the potential satisfy (V)n for some n 3. If 3H
has no zero-energy resonance or eigenvalue, then for any compact D IR
sup
j(x; k)j < 1
3
x2IR ;k2D
and for any with jj < n 2 there is a c < 1 such that
sup j@k (x; k)j < c (1 + jxj)jj :
k2Dnf0g
Proof. If H has no zero-energy resonance or eigenvalue the homogeneous
equation f = T0 f has no solution in B since, according to Lemma 2.7,
under the conditions (V)3 any solution of f = T0 f is a solution of ( 21 +
V (x))f (x) = 0 with f (x) = O(jxj 1). And a solution f 2 B of Hf = 0 with
f (x) = O(jxj 1) is, in particular, a resonance. Thus either f 2 L2 , i.e., zero
is an eigenvalue, or f 2= L2 , but then zero is a resonance.
Thus (1 Tk ) 1 exists for all k 2 D and, recalling (x; k) = ((1
Tk ) 1g)(x; k),
sup
j(x; k)j = sup kk kB sup k(1 Tk ) 1kL(B) kgk kB < 1 ;
3
x2IR ;k2D
k2D
k2D
since k(1 Tk ) 1kL(B) is a continuous function on a compact set and therefore
bounded. Recalling (x; k) = eikx + (x; k) the proof of the rst statement
is complete.
The bounds for the partial derivatives near zero also follow from the fact
that (1 T0 ) 1 exists if zero is neither a resonance nor an eigenvalue of
H . To see this we introduce spherical coordinates (jkj; !), jkj 2 (0; 1) and
! 2 1 for k. If we replace @ki jkj = ki=jkj = ! ei in Equation (2.16),
Equation (2.16) has a unique solution (; jkj; !) 2 B also for jkj = 0. Thus
limjkj!0 @k (x; jkj; !) exists for all ! 2 S 2. As in the rst part of this proof
@k (x; k) sup
sup
j (x; jkj; !)j < 1;
x2IR3 ;k2Dnf0g hxis+jj 1 x2IR3 ;jkj2[0;R];!2S 2
52
CHAPTER 2. MATHEMATICAL RESULTS
for some R such that D KR , follows from the fact that (; jkj; !) 2
B depends continuously on k. Noting j@k eikxj = jx1 1 x2 2 x3 3 eikxj < hxijj
completes the proof.
The uniform boundedness of as well as the bounds on its partial derivatives
with respect to k now follow from Proposition 2.10 and part (ii) of Theorem
2.4.
2.2 The ux-across-surfaces theorem
In this section we will prove rst the free ux-across-surfaces theorem and
then the one for short-range potentials. In Chapter 1 we saw that the uxacross-surfaces theorem is central to our derivation of scattering theory. Its
importance for quantum scattering was rst emphasized by Combes, Newton
and Shtokhamer in [9] almost 30 years ago. But although the ux is sometimes used in heuristical arguments in textbooks to motivate naive scattering
theory (see e.g. [27]), the rst attempts to prove the ux-across-surfaces theorem date back only a few years. The rst rigorous proof of the free case has
been given by Daumer et al. in [11] (1996). Short-range potentials were rst
treated rigorously by Amrein and Zuleta in [5] (1997) and a related result in
the case of long-range potentials has been established by Amrein and Pearson
in [3] (1997). For a short discussion of these results see Section 1.4.
2.2.1 The free ux-across-surfaces theorem
Although there are already two proofs of the free ux-across-surfaces theorem
([11], [5]) we will now give a third one for the following reasons: Both existing proofs are quite lengthy and complicated and partly cloud the relative
simplicity of the main idea. And since the result is of fundamental physical
importance it is certainly worthwhile to have a simple proof that can easily
be presented in textbooks or in lectures. Furthermore the proof provides us
with some control over the rate of convergence in (2.22) (see Remark 2.12).
Theorem 2.11. Let 0 2 S (IRn), the set of Schwartz functions, and
t := e iH0t 0. Then for all T 2 IR and any measurable 1
Z1 Z
Z1 Z
t
dt jj t (x) n(x)j d
lim
dt j (x) n(x) d = Rlim
R!1 T
!1 T
R
R
2.2. THE FLUX-ACROSS-SURFACES THEOREM
=
Z
C
53
jb 0(k)j2 dnk ;
(2.22)
where n(x) := x=jxj is the outward normal to R at the point x.
Proof. Since S is invariant under the free time evolution and since jb 0(k)j
doesn't depend upon t, it suces to show (2.22) for T 1:
Z1 Z
Z1 Z t
lim dt j (x) n d = Rlim
dt j t+T Te(x) n d
R!1
!1
T R
e R
2
Z T k2 e
Z 2
i
(
T
T
)
b
b k) dk :
= e 2
(k) dk = (
C
C
Therefore if (2.22) holds (for all 2 S ) for some xed T 1, then (2.22)
will hold for all T .
We proceed as in [11] and write for t 1 (see also Section 1.4)
Z ei jx 2tyj2
(y) dny
t (x) = e iH0 t0 (x) =
(2it) n2 0
x ei x22t Z xy y2 i x22t
e
b
+
=
e i t ei 2t 1 0(y) dny
(it) n2 0 t
(2it) n2
=: (x; t) + (x; t) :
(2.23)
The ux is now
x 2
x
n
t
j (x) = Im(t (x)rt (x)) = t t b 0 t + N (x; t) ;
(2.24)
with
x
x
n
1
b
b
N (x; t) := Im t 0 t r0 t + (x; t)r(x; t)
+(x; t)r (x; t) + (x; t)r (x; t)
:
(2.25)
We show as in [11] that the rst part of the ux in (2.24) gives rise to the
right hand side of (2.22):
x 2
Z1 Z
n 1 b 0 x n d
lim
dt
t
R!1 T
t
R! 2
Z 1R Z
n 1 b
= Rlim
dt
t
0 t Rn d
!1 T
R
Z R=T
Z
Z
n
1
2
b
= Rlim
djkjjkj
j0(k)j d
= jb 0(k)j2 dnk ;
!1
0
R
C
CHAPTER 2. MATHEMATICAL RESULTS
54
where d = Rn 1d
, d
being Lebesgue-measure on the unit sphere 1.
For the rst equality we introduced spherical coordinates x = R! and used
x n = R. For the second one we substituted k := xt = R!t , which, in
particular, means that djkj = Rt 2 dt. Since x n = jx nj, all equalities in
(2.22) hold, if we can show that
Z1 Z
lim
dt jN (x; t) nj d = 0 :
R!1 T
R
(2.26)
At this point we proceed dierently from [11]. We will show that there is a
c < 1 such that for every 0 < < 1
sup jN (x; t)j ct 1 R n+ :
(2.27)
x2R
Then (2.26) follows directly from this estimate:
Z1 Z
Z1
1+
lim
dt jN (x; t) nj d c Rlim
R
t 1 dt
R!1 T
!1
R
T
c
1+
= 0:
R
(2.28)
= Rlim
!1
T
Thus it remains to show (2.27), which follows from inserting the following
estimates for appropriate q into (2.25). Let kf kR := supx2R jf (x)j and
q 0, then there is a cq < 1, such that
q
q
b
b
0(=t)R cq Rt ;
jr0(=t)j R cq Rt
(2.29)
q
q
k(; t)kR cq t n2 Rt ;
k jr(; t)j kR cq t n2 Rt (2.30)
q
t q
n 1
n 1 t
2
2
k jr (; t)j kR cq t
:
k (; t)kR cq t
R ;
R (2.31)
The estimates (2.29) and (2.30) follow from the assumption that 0 2 S ,
i.e., 0, b 0 and their derivatives decay faster than any inverse polynomial.
We will prove (2.31) rst for q 2 IN0 by using
!q
!q
xy
xy
t
x
i
q
e t = i jxj jxj ry e i t
and doing q times integration by parts:
Z
!q "
!q
# 2 xy
y
n t
x
i
n
i
2t 1 0 (y ) d y j (x; t)j = (2t) 2 jxj
r
e
y e t
!q Z jxj !q y2 (2t) n2 jxt j jxxj ry ei 2t 1 0 (y) dny :
2.2. THE FLUX-ACROSS-SURFACES THEOREM
55
To conclude the bound in (2.31) on k kR for q 2 IN0 we are left to show that
the integral decays like t 1 , which follows from a simple calculation:
!
x ry q ei y22t 1 0 (y) =
jxj
L1
q
2
3
!
!
!q j
j y2 3 2
X q 4 x
x
i 2t 5 4
5
= jxj ry e
jxj ry 0 (y)
j=1 j
!q
y2 x
i
+ e 2t 1 jxj ry 0(y)
L1
X jj e X 2 ceq
t ky @y 0(y)kL1 + ctq
ky @y 0(y)kL1 :
:1jjq
:1j jq
:j j=q jj
Herein and denote multi-indices and we used that
y2 y2
ei 2t 1 :
2t
The case q 2 IR+0 follows from interpolation: For t=R 1 resp. t=R > 1,
(t=R)q is bounded by (t=R)n where n is the smallest integer larger than q
resp. the largest integer smaller than q.
Since also y0(y) 2 S , the second estimate in (2.31) can be computed in
the same way, doing, however, one more integration by parts.
Finally note that although the constants cq depend on q, they are bounded
uniformly in q when q varies only in some bounded subset of IR+0. Thus the
constant c appearing in (2.27) can be chosen independently of .5
Remark 2.12. It follows immediately from the proof that for any 0 2 S
and T 2 IR there is a C0;T such that for any > 0
Z 1 Z
C ;T 1
Z
2
b
dt jj t (x) n(x)j d
j
0(k)j T0 R1 :
T
R
C
Thus this proof provides us at least with some control on how fast the three
terms in (2.22) are approaching each other.
5 Actually it wouldn't change our proof at all if c in (2.27) would depend on , but it is
important for the conclusion of Remark 2.12.
56
CHAPTER 2. MATHEMATICAL RESULTS
2.2.2 The ux-across-surfaces theorem for short-range
potentials and wave functions without energy cutos
Now we come to the main result of this work, at least from the physical point
of view. We already saw in Section 1.4 that the ux-across-surfaces theorem
plays a central role for Bohmian and quantum scattering theory.
From a mathematical point of view the ux-across-surfaces theorem is
quite dierent in character from the main line of thought that dominated
mathematical scattering theory in the past decades. Most work has been
done on the question of asymptotic completeness (see Section 1.3), where
the Hilbert space structure of L2 and time asymptotics play a central role.
From a mathematical point of view these questions are very natural and
the results usually general and deep. However, as was argued in Chapter 1,
they provide only one step of many in the development of a physically sound
picture of scattering theory. As we shall explain, the mathematics necessary to prove the ux-across-surfaces theorem under physically reasonable
assumptions might dier from the usual strategies developed so far.
Recently the ux-across-surfaces theorem has been established for a large
class of short-range potentials by Amrein and Zuleta in [5]. However, since
they apply the usual time-dependent methods, they have to assume that
b out (k) has compact support not containing the origin. We argued in Section
1.4 that this assumption narrows the eld of applicability of the theorem and
seems unnecessary from a physical point of view.
We will shortly discuss the mathematical reason for demanding compact support for b out (k) away from the origin. The modern methods of
time-dependent scattering theory go back to Enss [17]6, who developed these
methods in order to prove asymptotic completeness of the wave operators.
This approach is based on information on how wave packets corresponding
to scattering states move in space for large times. It is essential to prove
so-called propagation estimates having, for example, the form
khxi e iHt f (H )hxi k c(1 + jtj) + ;
where f 2 C01(IR+) provides the energy cuto, > 0 and is a positive
constant depending on the decay of the potential. There are several dierent
methods of proving this or similar estimates. However, at some point they
all rely on integration by parts in Fourier representation in a way often called
stationary phase method.7 It works for a dense set of wave functions, namely
6 For a comprehensive treatment see, e.g., [22]
7 In the proof of Theorem 2.20 we use a Theorem going back to Hormander that is basic
to the proof of asymptotic completeness by the Enss method as presented in [22].
2.2. THE FLUX-ACROSS-SURFACES THEOREM
57
for those 0 such that b out (k) is compactly supported away from the origin,
since k = 0 would be a point of stationary phase. In the context of asymptotic
completeness, however, it is sucient to prove these estimates on a dense
subset of Hac since density-in-L2 arguments can be applied easyly in the
end.
Remark 2.13. This is exactly the way we will go in Section 2.3. There
we rst show that some sequence of uniformly bounded operators strongly
approaches zero on an open set of vectors, dened by certain energy cutos.
Then we will conclude that this holds true also on the closure of this set.
Amrein and Zuleta show that one can apply these \time-dependent methods" to prove also the ux-across-surfaces theorem. But in this case it is not
sucient to prove the theorem on a dense set of wave functions, since the
set of wave function for which (1.31) holds can not be enlarged by a simple
limiting procedure in L2 . This is because the expression
Z1 Z
dt j t n d ;
0
R
giving the time intergrated ux through a surface at nite distance, is an
unbounded sesquilinear form and thus
Z1 Z
Z1 Z
n
t
t
nlim
!1 0 dt R j n d = 0 dt R j n d
can not be concluded from limn!1 knt t k = 0. Hence in order to enlarge
the set of admissible wave functions on has to prove (1.31) directly for this
larger class of wave functions.
The rst idea would be to use propagation estimates for wave functions
without energy cutos that have been already established in the literature
(see [21], [29]). But it turns out that these estimates are not sucient to
prove (1.31) directly without further highly nontrivial input. On the other
hand these estimates are in some sense much stronger than necessary for our
purpose and were only established for a very restricted class of potentials.
Hence we will prove (1.31) directly for a class of wave functions without
energy cutos using the results on generalized eigenfunctions we established
in the Section 2.1.
Theorem 2.14.
Let the potential satisfy the condition (V)4 and let
zero be neither a resonance nor an eigenvalue of H . Let out 2 S . Then
t = e iHt W+out is continuously dierentiable except at the singularities
of V and for any measurable S1 and any T 2 IR
CHAPTER 2. MATHEMATICAL RESULTS
58
Z1 Z
Z1 Z
t
lim dt j (x) n(x) d = Rlim
dt jj t (x) n(x)j d
R!1
!1
T
R
Z T R
= jb out(k)j2 dk :
(2.32)
C
Remark 2.15.
Due to the intertwining property of the wave operators
(1.19) and the fact that S is left invariant under the free time evolution,
the condition imposed on in Theorem 2.14 is invariant under the full time
evolution: e iHt W+S = W+e iH0 t S = W+S .
It would be of course more satisfactory if we could prove (2.32) under
a suitably general condition on 0, not on out . However, the set of wave
functions 0 = W+ out for which Theorem 2.14 holds is dense in Hac(H )
since S is dense in L2 and W+ : L2 ! Hac(H ) is unitary. For an explicit
characterization of the domain W+S one needs suitable mapping properties
of the wave operators. Since we explicitly aimed at a result avoiding cutos
at zero energy, most existing results on mapping properties of the wave operators (see, e.g., [13]) are of no help to us. Some mapping properties for wave
functions without energy cutos have been established in [29], but they are
not sucient for our purpose.
In the following corollary we will establish sucient conditions on 0
such that out 2 S , albeit for a much smaller class of potentials than those
allowed in Theorem 3.14.
Corollary 2.16. Let the potential satisfy (V)1 and let zero be neither a
resonance nor an eigenvalue of H . Let 0 2 Ran(EI (H )) such that hxiN 0 2
L1 for all N 2 IN. Here EI (H ) denotes the spectral projection of H on some
bounded interval I IR with I \ pp (H ) = ;.
Then out = W+ 10 2 S and thus (2.32) holds.
Proof. We will show that b out 2 C01 S and therefore also out 2 S .
First recall that, according to Proposition 2.1.(v), b out = F+0. By
assumption we have that hxiN 0 2 L1 for all N 2 IN. Hence b out 2 C 1 by
virtue of Corollary 2.6.
To see that b out has compact support note that, by assumption, there
is a function f 2 C01(IR) such that f (H )0 = 0 . Using the intertwining
relations (1.20) we compute
f (H0)out = f (H0)W+ 10 = W+ 1f (H )0 = W+ 10 = out :
2.2. THE FLUX-ACROSS-SURFACES THEOREM
59
Hence f (jkj2=2)b out (k) = b out (k) and suppb out suppf follows.
Remark 2.17. To see that the set of wave functions specied in Corollary
2.16 is not empty|indeed it is dense in Hac (H )|observe that for (V )1
potentials and functions f 2 C01(IR)
hxiq f (H )hxi q
is a bounded operator on L2 (IR3 ) for every q 0 (see Lemma 3 in [4]).
Hence take any 2 L2 (IR3 ) such that hxiN 2 L2 for all N 2 IN. Then for
0 := f (H ) we have 0 2 RanEI (H ) with I = suppf and
hxiN 0 = hxiN f (H )hxi
N
hxiN 2 L2
for all N 2 IN and thus, by Holder's inequality, also hxiN 0 2 L1 (IR3).
Remark 2.18.
In [5] and [3] the ux-across-surfaces theorem is as well
proved under conditions imposed on out and then mapping properties of the
wave operators for wave functions with energy cutos are applied in order to
get sucient conditions on 0. Also in this case the original result holds for
a larger class of potentials then the one involving conditions on 0 .
Proof [of Theorem 2.14].
Let t = e iHt W+out , out 2 S . With
Proposition 2.1.(iv), (v) and (x; k) := + (x; k) eikx we have that
1 Ze
(2) 23
Z
= 13 e
(2) 2
t(x) =
i k22 t b
out (k )+ (x; k ) dk
Z
ikx dk + 1
(
k
)
e
e
out
(2) 32
i k22 t b
i k22 t b
out (k ) (x; k ) dk
=: (x; t) + (x; t) :
(2.33)
The ux generated by this wave function is
j t (x) = Im(r + r + r + r ) ;
(2.34)
where the dierentiability of is obvious and that of will be established
later.
CHAPTER 2. MATHEMATICAL RESULTS
60
The rst part j0 = Im(r) is the ux generated by the free time evolution of out and according to the free ux-across-surfaces theorem (Theorem
2.11)
Z1 Z
Z1 Z
Z
lim
dt
j
(
x;
t
)
n
d
=
lim
dt
j
j
(
x;
t
)
n
j
d
=
jb out(k)j2 dk :
0
0
R!1
R!1
T
R
T
R
C
Therefore to prove (2.32) we need only show that the last three terms in
(2.34) do not contribute to the ux across distant surfaces, i.e., that for
j1 := Im(r + r + r )
Z1 Z
lim dt jj1(x; t) nj d = 0 :
R!1
T
R
(2.35)
For some xed T > 0 this will follow from the estimates (which we shall
prove below)
sup j(x; t)j t 23 f1(R; t) 8t T
(2.36)
3
(2.37)
x2R
sup jr(x; t)j t 2 f2(R; t) 8t T ;
x2R
where there exists a c < 1 such that fi(R; t) satisfy
and
8t T
(2.38)
fi(R; t) < c
(2.39)
lim f (R; t) = 0
R!1 i
sup
R2[0;1);tT
for i = 1; 2, and there is R0 0 such that
1 ; 8R > 0 ;
R(t + R)
x2R
sup jr (x; t)j c 1 ; 8R > R0
R(t + R)
x2R
sup j (x; t)j c
for t T . Note that the constants in these estimates depend on T .
Using (2.36) and (2.41) we obtain by dominated convergence
(2.40)
(2.41)
2.2. THE FLUX-ACROSS-SURFACES THEOREM
lim
R!1
Z1 Z
T R
jIm(r ) nj d dt
61
Z1
Rlim
4 sup R2 jjjr j dt
!1
T x2R
Z1 R2f1 (R; t)
c Rlim
dt
!1 t 32 R(t + R)
T
= c
Z1
T
Rf1(R; t) dt = 0 (2.42)
lim
R!1 t 32 (t + R)
for T > 0 where we observed that the integrand in (2.42) is bounded by an
integrable function uniformly in R,
Rf1 (R; t) ct 32 :
t 32 (t + R)
The terms j rj and j r j can be treated analogously and thus (2.35)
holds for positive times T .
According to Remark 2.15, the set of wave functions for which (2.32)
holds as well as the right hand side of (2.32) are invariant under nite time
shifts:
Z1 Z
Z1 Z t
lim dt j (x) n d = Rlim
dt j t+T Te(x) n d
R!1
!1
T R
e R
2
Z T k2 e
Z
i
(
T
T
)
b
= e 2
out (k) dk =
C
C
b
2
out (k) dk :
Therefore if (2.32) holds (for all out 2 S ) for some xed T , then (2.32) will
hold for all T ; hence (2.32) is proved for all T .
We turn now to the proof of the estimates (2.36-2.41). Recalling that
(x; t) = (e iH0 tout )(x) and, since r commutes with the free time evolution,
r(x; t) = (e iH0 trout )(x), we can write
Z jx yj2
1
ei 2t out (y) dy
(x; t) =
(2.43)
(2it) 32
and
Z jx yj2
1
r(x; t) =
ei 2t rout (y) dy :
(2it) 32
(2.44)
CHAPTER 2. MATHEMATICAL RESULTS
62
(2.36-2.39) are now immediate consequences of (2.43) and (2.44) and the fact
that, for every xed t T , (x; t) and r(x;Rt) are Schwartz functions.
By (2.2) (x; k) = +(x; k) eikx = 21 e jixjkjjxyj yj V (y)+(y; k) dy and
therefore
(x; t) =
=
=
=:
1 Z e i k22t b (k)(x; k) dk
out
(2) 32
"
#
1 Z e i k22 t b (k) Z e ijkjjx yj V (y) (y; k) dy dk
out
+
jx yj
(2) 52
1 Z V (y) Z e i( k22 t +jkjjx yj)b (k) (y; k) dk dy
out
+
(2) 52 jx yj
1 Z V (y) f (x; y; t) dy
(2.45)
(2) 52 jx yj
where
Z
2
f (x; y; t) := e i( k2 t +jkjjx yj)b out (k)+ (y; k) dk :
(2.46)
The change of order of integration in (2.45) is justied by Fubini's theorem.
We shall now apply \stationary phase" methods to estimate (2.46). We set
k2 t + jkjjx y j
:= 2 t + jx yj
2
1
d
0 = djkj = jktjt++jxjx yyj j = jk1 j++jjxx yyjjtt 1 min(1; 2jkj)
2
2
t
! := 2 + jx yj:
In the following 0 will denote dierentiation with respect to jkj. Introducing
spherical coordinates, with d
denoting Lebesgue measure on the unit sphere,
we estimate (2.46):
Z
"
#
1
d
i!
2
b
out (k)+(y; k)jkj djkj d
(k)
jf (x; y; t)j = !0 djkj e
Z
"
#
1
d
1
i!
2
= ! e djkj 0 b out (k)+(y; k)jkj djkj d
(k)
#
Z d " 1
1
2
b
! djkj 0 out (k)+ (y; k)jkj djkj d
(k) : (2.47)
For the second equality in (2.47) 0the boundary term from the partial integration at jkj = 1 vanishes since 1 max(1; 2j1kj ), limjkj!1 jkj2b out (k) = 0,
2.2. THE FLUX-ACROSS-SURFACES THEOREM
63
and + is bounded according to Proposition 2.1.(vi). The boundary term at
jkj = 0 vanishes since b out and + are bounded and 0 1jkj2 max(jkj2; jk2j ).
Note that the dierentiability of + is ensured by Theorem 3.4.(i). Next,
observe that
"
#
d 1 b out + jkj2 10 00b out + jkj2 + 1 b 0 +jkj2
djkj 0
2
0 out
+ 10 b out 0+jkj2 + 10 b out +2jkj : (2.48)
Since 00 = ( 12 + jx yjt 1) 1 2, we obtain for the rst term
Z 1
00
b
0 2 out + dk 0
1
Z
Z
B jb out(k)j
C
sup 3 j+(y; k)j B
2jb out(k)j dkC
dk
+
@
A c1 :
2
2jkj
y;k2IR
1
1
jkj<
jkj
2
2
(2.49)
Analogously we get for the second and fourth term in (2.48)
Z 1
Z 2
0
0 b out + dk c2 and 0 jkj b out + dk c4
By By Theorem 2.4.(iii) the third term satises a bound linear in jyj:
Z 1
0 b out 0+ dk ce3 sup
j0+(y; k)j c3 (1 + jyj) :
3
k2IR nf0g
Combining the four estimates we arrive at
jf (x; y; t)j c !1 (1 + jyj) = c t +1 +jxjyj yj ;
2
which inserted into (2.45) yields
Z V (y)
1
sup j (x; t)j = sup 5 jx yj f (x; y; t) dy
x2R
x2R (2 ) 2
Z
c sup jx jV y(yj()tj(1++jxjyj)yj) dy :
x2R
2
(2.50)
(2.51)
CHAPTER 2. MATHEMATICAL RESULTS
64
Now, substituting z = x y,
Z
jV (y)j(1 + jyj) dy = Z jV (x z)j(1 + jx zj) dz
jx yj( 2t + jx yj)
jzj( 2t + jzj)
j
x
j
jzj< 2
Z
jV (y)j(1 + jyj) dy
+
jx yj( 2t + jx yj)
jxj
jx yj 2
Zjx2j 4jzj2
sup jV (x z)j(1 + jx zj) jzj( t + jzj) djzj
z2B jxj
2
0
2
Z
+ jxj t 1 jxj jV (y)j(1 + jyj) dy ;
2 (2
+ 2)
where Br denotes the ball with radius r in IR3 centered at the origin. Since
V (x) = O(jxj 4 ) for some > 0,
sup jV (x z)j(1 + jx zj) cjxj
z2B jxj
2
3
for jxj suciently large. Using
t t
2
Z z
dz = + t ln t + + t t + 1 = t + 0 t+z
we compute
Zjx2j
0
jzj2
1 jxj2 :
d
j
z
j
jzj( t + jzj)
2 t + jxj
2
R
Finally jV (y)j(1 + jyj) dy < 1 so that altogether
jxj2 +
1
sup j (x; t)j c sup jxj t +
jxj jxj(t + jxj)
x2R
x2R
3
!
c :
= R(t +
(2.52)
R)
We now show that the same bound holds for supx2R jr (x; t)j and R > R0,
where R0 is chosen such that all singularities of V lie inside of the ball with
radius R0 . Then
2.2. THE FLUX-ACROSS-SURFACES THEOREM
65
Z V (y) Z i( k2 t +jkjjx yj) b
1
2
jr (x; t)j (
k
)
(
y;
k
)
dk
dy
e
out
+
5 (2) 2 jx yj2
Z
Z k2 t
1
V
(
y
)
i
(
+
j
k
jj
x
y
j
)
2
+ 5 e
j
k
j
b out(k)+(y; k) dk dy ;
(2) 2 jx yj
(2.53)
where the exchange of dierentiation and integration will be justied below. The second term can be treated analogously to j (x; t)j, since also
jkjb out(k) 2 S . The rst term can as well be estimated along the same lines:
in Equation (2.51) jx yj2 will appear in the denominator instead of jx yj,
which leads to a stronger bound than (2.52).
To get (2.53) from (2.45) we note that according to Proposition 2.1 (; k)
is a classical solution of the stationary Schrodinger equation. Thus (; k) as
well as (; k) are dierentiable with respect to x except at the singularities
of V . We will show that
!
Z ijkjjx yj
rx(x; k) = rx 21 ejx yj V (y)+(y; k) dy
Z
ijkjjx yj !
1
e
= 2 rx jx yj V (y)+(y; k) dy
and that therefore
jrx(x; k)j c1 + c2 jkj
(2.54)
(2.55)
for some c1 ; c2 < 1. Then changing the order of dierentiation and integration in the rst line of (2.45) is justied by dominated convergence and
(2.53) follows for all x which are not singularities of V .
To get (2.54) for some x0 2 IR3 which is no singularity of V , we split the
domain of integration into B2R (x0 ) := fy 2 IR3 : jx0 yj 2Rg and its complement B2cR (x0 ), where R is chosen such that B2R (x0 ) contains no singularity
of V . Then one can change the order of integration and dierentiation in the
B2cR (x0 ) term, since there the integrand is bounded by an integrable function
uniformly in x for x 2 BR (x0 ). To see that the B2R (x0 ) term can be made
arbitrary small by appropriately choosing R, we write down the dierence
quotient for this term. Using that supy2B2R (x0 ) (V (y)+(y; k)) ck < 1 and
that jrei (r + r)ei(+)j jrj + jrj we compute
CHAPTER 2. MATHEMATICAL RESULTS
66
!
Z
i
j
k
jj
x
+
y
j
i
j
k
jj
x
y
j
1 e
e
lim
V
(
y
)
(
y;
k
)
dy
+
jj!0 jj jx + yj jx yj
B2R (x0 )
ijkjjx+ yj
Z
jx yj e ijkjjx yjjx + yj dy
1c
e
jlim
k
j!0 jj
jx + yjjx yj
B2R (x0 )
Z jx yjjkjjj + jj
1
c
dy
jlim
j!0 jj k
j
x
+
y
jj
x
y
j
B2R (x0 )
!
Z
j
k
j
1
jlim
c
+ jx + yjjx yj dy ck 12R ;
j!0 k
j
x
+
y
j
B (x0 )
2R
where the last inequality is justied by elementary integrations. The bound
(2.55) can be obtained by a simple calculation.
From that we also conclude that t (x) is dierentiable outside the singularities of V (x) since and are.
2.3 Approximate eigenstates of W
In this section we will conrm the heuristic insight that any wave function
2 L2 (IR3 ) supported far away from the scattering center containing only
components moving toward the latter is an approximate eigenstate of the
wave operator W , i.e., that k(W 1)k 0.
Before we can state and proof our result we introduce the so-called Enss
condition on the potential V (see [22] Denition 4.1):
Denition 2.19. The potential V : IR3 ! IR is said to satisfy the Enss
condition if
(i) V is H0-bounded with relative bound smaller than 1.
(ii) The bounded, monotone decreasing function
h(R) = kV (H0 + i) 1(jxj R)k
is integrable on (0; 1), where denotes the characteristic function.
It was shown by Enss that potentials satisfying his condition are asymptotically complete (see [22] for a detailed presentation and references). Note
that potentials satisfying (V)2 also satisfy the Enss condition.
2.3. APPROXIMATE EIGENSTATES OF W
67
We will now proof the following theorem:
Theorem 2.20. Let V satisfy the Enss condition and let W be the wave
operator associated with H = H0 + V dened as the operator sum. Let 0
be such that b 0(k) is supported in a half space P := fk 2 IR3 : k 0g
for some 2 1.8 If n(x) := (T n0)(x) := 0 (x + n) denotes the
translation of 0 along the vector n, n 2 IN, then
nlim
!1 k(W
1)nk = 0 :
(2.56)
Remark 2.21.
It is well known that (2.56) holds for all 0 2 Hac(H )
if T n is replaced by einH , i.e. if the wave function is evolved backwards in
time (see, e.g., [22] pp. 86). Both problems are clearly related and we will
make use of several tools and results from time-dependent scattering theory
that are also applied in the proof of the result mentioned above.
First we introduce the symmetric dilation operator D := 12 (x p + p x),
p := ir, dened on C01(IR3nf0g). Heuristically D probes the angle between
the position and the momentum of a particle and thus discriminates incoming
and outgoing states. D has a unique self adjoint extension (also denoted by
D) that is diagonalized by the unitary map M : L2 (IR3) ! L2 (IR) L2 (1 )
given by
1 Z N jxj 12 i(jxj; !)djxj ;
(M)(; !) = s Nlim
(2.57)
!1 2 N1
where again x = !jxj. For 2 C01(IR3 nf0g) the formula holds pointwise and
F DF 1 = D. Furthermore D has purely absolutely continuous spectrum
on ( 1; 1) and we denote the projections on its positive resp. negative
spectral subspaces by P+ and P . For a proof of these results see e.g. [22]
Proposition 6.2. We will also need a Lemma which is central to Enss' proof
of asymptotic completeness (see [22] Lemma 7.6'):
Lemma 2.22. Let n be a sequence of vectors with the property that
limn!1 k(jxj < n=2)nk = 0 and EI (H )n = n , where EI (H ) is the
projection on the spectral subspace of H for some bounded subinterval I of
IR+ not containing zero. Let n;in = f (H0)P n and n;out = f (H0)P+n
where f 2 C01(IR+) satises f = 1 on I . Then:
(i) limn!1 kn n;in n;out k = 0:
8 Recall that 1 denotes the unit sphere in IR3 .
68
CHAPTER 2. MATHEMATICAL RESULTS
(ii) limn!1 k(W+ 1)n;out k = 0 and limn!1 k(W
1)n;ink = 0.
Thus every sequence of vectors leaving any bounded region in space and
staying in a nite energy interval splits into outcoming and incoming components that become approximate eigenvectors of the wave operators. These
components are given by the projections onto the positive and negative subspaces of D.
Proof [of Theorem 2.20]. For the beginning we assume that b 0(k) 2
C01(IR3 nf0g) is supported in a cone C; such that k=jkj cos for some
< and all k 2 suppb 0(k). Let f 2 C01(IR nf0g) be a real valued function
such that f (jkj2=2) = 1 whenever k 2 suppb 0, i.e., f (H0)0 = 0 .
Let n(x) := 0(x+n), then the sequence n weakly goes to 0, since it is
just the translation of 0(x) along the vector n. We dene e n := f (H )n
and write
(W 1)n = (W 1)f (H0 )n = (W 1)e n +(W 1)(f (H0 ) f (H ))n :
It is well known that for our class of Hamiltonians the operator f (H0) f (H )
is compact for f 2 C1(IR) (see e.g. [22] Lemma 7.3). Since w limn!1 n =
0 and since (W 1) is bounded, the compactness of f (H0) f (H ) implies
that limn!1 k(W 1)(f (H0) f (H ))nk = 0. We will now apply Lemma
2.22 to show that also limn!1 k(W 1)e nk = 0. Therefore we note that
the condition on the support of the spectral measure of e n is satised by
construction. To see that the sequence e n also satises limn!1 k(jxj <
n=2)e nk = 0 we do the same splitting as before:
(jxj < n2 )e n = (jxj < n2 )f (H )n =
(jxj < n2 )f (H0)n + (jxj < n2 )(f (H ) f (H0))n :
Now the norm of the rst term approaches zero since H0 commutes with
translations and for the norm of the second we us again the compactness of
f (H ) f (H0) and the weak convergence of n to zero. Hence Lemma 2.22
implies that
1)e nk nlim
1)e n;ink + nlim
1)e n;out k
nlim
!1 k(W
!1 k(W
!1 k(W
= nlim
1)e n;outk
!1 k(W
= nlim
1)f (H0)P+e nk ;
!1 k(W
where 3.22 (i) was used for the rst and (ii) for the second (in-)equality. Since
(W 1)f (H0) is a bounded operator we would be done if we could show that
2.3. APPROXIMATE EIGENSTATES OF W
69
the norm P+e n or, equivalently|using the compactness of f (H0) f (H ) for
a third time| that the norm of P+n approaches zero. Heuristically this is
clear, because we constructed n such that it contains no outgoing components. For the proof we use the unitary transformation M that diagonalizes
D. With
Z1 1
1
gn(; !) := 2 jkj 2 i b n(jkj; !)djkj
(2.58)
0
we have
Z Z1
kP+nk = 0 jgn(; !)j2 d d! :
1
We will show below that there is a c < 1 such that
jgn(; !)j 1 + nc + (2.59)
and thus
kP+nk 14c
:
+n
2
Hence limn!1 kP+nk = 0.
In order to get the estimate for jgn(; !)j we note that
b n(k) = e ink b 0 (k)
and rewrite
Z1
gn(; !) = 21 jkj 21 i e inkb 0 (jkj; !)djkj
Z0 1
1
= 2 e i( ln jkj+njkj cos )e 12 ln jkjb 0(jkj; !) djkj
Z0 1
1
=:
e in (jkj;!;)u(jkj; !) djkj ;
(2.60)
2 0
where
we abbreviated cos (!) := k=jkj, := 1 + + n, u(jkj; !) :=
1 ln jkj b
e 2 0(jkj; !) and
(!) :
n (jkj; !; ) := ln jkj1++njk+j cos
n
Since for k 2 suppb 0 we have that cos (!) > cos > 0 and 0 < jkjmin <
jkj < jkjmax < 1, the following estimates hold uniformly for k 2 suppb 0 ,
n 1, 2 [0; 1) and ! 2 1:
(i)
@ (jkj; !; ) = n cos (!) + =jkj 1 min(cos ; jkj 1 ) :
max
@ jk j n
1+n+
2
70
(ii)
(iii)
CHAPTER 2. MATHEMATICAL RESULTS
2
1 < 1 :
@ n (jkj; !; ) = @ jkj2
1 + n + jkj jkjmin
@ u(jkj; !) < ce < 1 :
@ jkj
Thus the oscillating integrand in (2.60) has no stationary phase and we can
apply Hormander's theorem (see e.g. [22] Theorem 2.2) to conclude (2.59).
This concludes the proof for b 0(k) compactly supported in a cone with
opening angle smaller than and not containing the origin. But we can
use the fact that (W 1)T n are uniformly bounded to conclude that
limn!1 k(W 1)T n0k = 0 for any 0 in the closure of this set, which
is, quite obviously, the set of 0 2 L2 (IR3 ) such that b 0 (k) is supported in
the half space P.
Remark 2.23. Of course the symmetric result for n(x) := 0 (x n)
and W+ instead of W is true by the same proof:
nlim
!1 k(W+
1)Tn0k = 0 :
c k )j2 ! (k )
2.4 The limit j(
0
In this section we will nally show that for wave packets with momentum
support narrowly peaked at some k0 formula (1.40) for the cross section
measure applies.
We start with an immediate corollary of Theorem 2.4 about the regularity
of the T -matrix (see also Section 1.5):
Corollary 2.24. Let V satisfy (V)n for some n 3 and let zero be neither
a resonance nor an eigenvalue of H . Then the T -matrix T (; ) dened by
T (k; k0) = (2)
3
Z
e
ikxV (x)
(x; k0 ) dx
is a bounded and continuous function on IR3 IR3. Furthermore
(2.61)
b K )j2 ! (K 0 )
2.4. THE LIMIT j(
71
(i) T (; ) 2 C n 3(IR3 (IR3 n f0g))
(ii) For every multi-index with jj n 3
sup
3 0
k2IR ;k 2IR3
and
j@k T (k; k0)j < 1
sup
j@k0 T (k; k0)j < 1 :
3
0
3
k2IR ;k 2IR nf0g
Proof. Since in particular V 2 L1 (IR3 ) whenever V satises (V)3 and since
is uniformly bounded and continuous, dominated convergence implies
that T is uniformly bounded and continuous in both variables.
(i) and (ii) follow by changing the order of integration and dierentiation{
also justied by dominated convergence|and using the bounds from Theorem 2.4.
It was shown, e.g. in [26], Theorem XI.42, that for f; g 2 C01(IR n f0g)
Z
(f; (S 1)g) =: (f; Tg) = i f (k)
holds.9 Since in our case the function
h(k) := i
Z
1
Z
1
T (k; !0jkj)g(!0jkj)jkj d
0 dk
(2.62)
T (k; !0jkj)g(!0jkj) ; jkj d
0
(2.63)
is well dened (where still g 2 C01(IR3 n f0g) is assumed) and in L2 , from
(f; Tg) = (f; h) for a dense set of vectors it follows that Tg = h in the
L2 -sense.
Hence for in such that b in 2 C01(IR n f0g) we have
Sd
in(k) = b in(k) i
Z
1
T (k; !0jkj)b in(!0jkj)jkj d
0 :
(2.64)
If we further assume that b in(k) is supported in some cone C0 the formula
(1.36) for the scattering cross section can be evaluated for solid angles 1
with C \ C0 = f0g solely in terms of T :
in () = 2
2
Z Z
T (k; !0jkj)b in(!0jkj)jkj d
0 dk :
C 1
(2.65)
9 Note that in [26] a slightly weaker statement was proven. But for our class of potentials
their proof implies the stated result, because the \exceptional set" is empty in our case.
CHAPTER 2. MATHEMATICAL RESULTS
72
Now, as explained in Section 1.6, we will evaluate this expression for initial
wave functions with momentum support only in a small neighborhood of
some k0 . More precisely we will aim for an expression for depending only
on k0 by taking the limit jb in(k)j2 ! (k k0 ).
Therefore let f 2 C01([ 1; 1]) be a real valued, positive function with
kf kL1 = 1. Then the sequence of functions nfn (t) := nf (nt), n 2 IN,
approaches (t) in the sense that for any continuous function g(t)
nlim
!1
Z
g(t)nfn(t) dt = g(0) :
(2.66)
Recall how this follows from the mean-value theorem: Since nfn (t) 0 8t
and g(t) is continuous, there is a n 2 [ n1 ; n1 ] such that
Z
Z
g(t)nfn(t) dt = g(n) nfn (t) dt = g(n) ;
which immediately implies (2.66). In the same sense we also have that
1
nf
n (t) 2
nlim
!1 R f (t) 12 dt = (t)
(2.67)
and for x; x0 2 IR3, using \spherical" coordinates x = (jxj; cos ; ),
n3 fn(jxj jx0j)fn(cos cos 0 )fn( 0 ) = (x x )
lim
0
n!1
jxj2
is a regularization of the 3-dimensional delta function. In order to get
jb n(k)j2 ! (k k0) we dene
3
cos 0 )fn( 0)
b n(k) = pn(k) n fn(jkj jk0j)fn(cos
jkj2
! 12
; (2.68)
where pn(k) is an arbitrary phase, i.e., jpn(k)j = 1. Now we are ready to
formulate and proof the following
Theorem 2.25. Let V satisfy (V)3 such that zero is neither a resonance
nor an eigenvalue of H and let b n be a sequence in L2 (IR3 ) of the form (2.68).
Then for measurable 1
k0 () :=
R jTd (k)j2 dk R
0
2 0
n
C
jT (! jk0 j; k0 )j d
R
lim
=
R
n!1 3 jTd
n(k)j2 dk 1 jT (!0jk0j; k0)j2 d
0
IR
(2.69)
b K )j2 ! (K 0 )
2.4. THE LIMIT j(
73
Proof [of Theorem 2.25]. First note that Corollary 2.24 assures that
T (k; k0) is a continuous and bounded function on IR3 IR3 . Hence the following calculations are all rigorous. According to (2.64) we have
Td
n(k) =
=
i
Z
T (k; !0jkj)b n(!0jkj)jkj d
0
1Z
jk0j) 2 T (k; !0jkj)pn(!0jkj)fn(cos 0 cos 0 ) 12
1
in 32 fn(jkj
fn
(0
1
1
0) 2 d(cos 0)d0 ;
and thus
Z
Z
2
d
jT n(k)j dk = 3 2 fn(jkj jk0j) 3
IR
ZIR
T (k; !0jkj)pn(!0jkj)n 43 fn(cos 0 cos 0 ) 12 n 34 fn(0 0) 12 d(cos 0)d02 dk
1
The expression j : : : j2 vanishes pointwise for each k in the
limit n ! 1,
3
since both of the functions fn inside carry only a factor n 4 . Thus the whole
integrand vanishes pointwise and we have
nlim
!1 kT nk = 0 :
This was expected because if the wave function spreads out in space fewer
particles are scattered. In order to get a nite expression one has to take
the limit for the probability that a particle is scattered into the cone C
conditioned by the event that it is scattered at all. By a simple computation
we have
R jTd
n(k)j2
=
RC jTd
2
IR3 n (k )j
R f (jkj jk j) R T (k; !0jkj)p (!0jkj)n 43 f (cos 0 cos ) 21
n
0
n
n
0
R1
= RC
3
1 0
0
0
IR3 fn (jkj jk0 j) 1 T (k; ! jk j)pn(! jk j)n 4 fn (cos cos 0 ) 2
2
n 43 fn(0 0) 12 d(cos 0 )d0 dk
3 0
2
n 4 fn( 0) 12 d(cos 0 )d0 dk
=
R nf (jkj jk j) R T (k; !0jkj)p (!0jkj)nf (cos 0 cos ) 12
n
0
n
n
0
C
R nf (jkj jk j) R 1 T (k; !0jkj)p (!0jkj)nf (cos 0 cos ) 12 0
n
n
0
IR3 n
1
2
nfn(0 0) 21 d(cos 0)d0 dk
2
nfn(0 0) 12 d(cos 0)d0 dk
CHAPTER 2. MATHEMATICAL RESULTS
74
=
=
R nf (jkj jk j)j jT (k; ! (k)jkj)j2 jkj2 djkj d
RC nfn(jkj jk0j)j jT (k; !n(k)jkj)j2 jkj2 djkj d
0
n
IR3 n
R jT (! (!); ! (!) (!))j2 (!)2 d
R jT (!n (!); !n (!)n (!))j2 n (!)2 d
:
n
n
n
n
1
For the second equality we just multiplied the numerator and the denominator both with n2 . For the third equality we use (2.67) together with the
mean-value theorem that ensures the existence of the appropriate continuous functions !n(k) = (n(k); n(k)) with j cos n(k) cos 0 j 1=n and
jn(k) 0 j 1=n and, for the last equality, that of appropriate n(!) with
jn(!) jk0jj 1=n. Recalling that T is continuous, we nally conclude that
R jTd
R jT (! ; ! )j2 d
R jT (!jk j; k )j2 d
n(k)j2
n n n
0 0
C
=
lim
=
;
R
R
2
2
nlim
!1 R 3 jTd
n
!1
n(k)j2
1 jT (!n; !n n )j d
1 jT (! jk0 j; k0 )j d
IR
where the order of the limit and the integration can be changed by virtue of
dominated convergence and the fact that T is bounded.
Remark 2.26. We had to perform two dierent limits in order to arrive
at formula (2.69) for the scattering cross section: First, in was spatially
moved out to innity to assure that k(W 1)ink 0 and thus allow for
the replacement of W by S . Second, jb in(k)j2 has to be sharply peaked at
some k0 in order to get a formula only depending on k0 and not on the shape
of in. At rst sight one might worry that these two limits are incompatible,
because peaking in momentum space means spreading out in position space
and thus, possibly, destroying the eects of the rst limit.
However, it is easy to see that one can do both limits at the same time: In
the wave function (2.68) we allowed for an arbitrary phase pn(k), which could
be used to move out the wave packet in position space suciently fast without
changing the proof. For such a sequence limn!1 k(W 1)nk = 0 can be
shown by exactly the proof of Theorem 2.20: According to the Theorem of
Hormander we use there, the constant c in (2.59) depends on b 0 only via the
factor
b b
d
0 1;1 := sup 0(jkj; !) + sup djkj b 0 (jkj; !) :
k
k
One easyly calculates that for b n(k) =: pn(k)b n;0 dened by (2.68) there is
a c < 1 such that
b n;01;1 cn4 :
2.5. SOME TECHNICAL LEMMAS
75
Hence, chosing for the phase e.g. pn(k) = e in5kk0=jk0j, one gets instead of
(2.59)
4
jgn(; !)j 1 + cn
n5 + ;
which is still sucient to conclude limn!1 k(W 1)nk = 0.
2.5 Some technical lemmas
In this section we will prove two technical lemmas used in the proof of Theorem 2.4.
Lemma 2.27. Let I be a compact subset of IR3 and
Be := ff (x; k) 2 C (IR3 I ) : jxlim
sup jf (x; k)j = 0g :
j!1
k2I
(i) The space Be equipped with the norm
kf kBe = sup
jf (x; k)j
3
x2IR ;k2I
is a Banach space.
(ii) Let fAk gk2I L(B ) be a family of bounded operators on B := C1(IR3)
such that Ak depends continuously on k with respect to the operatornorm. Then for f 2 Be
(Af )(x; k) := (Ak f (; k)) (x)
denes an operator A 2 L(Be ) and kAkL(Be) supk2I kAk kL(B) .
Proof [of part (i)]. Let3ffngn2IN be a Cauchy sequence in Be C (IR3 I ).
Then there exists f 2 C (IR I ) such that limn!1 kf fn kBe = 0. It remains
to show that f 2 Be , i.e., that limjxj!1 supk2I jf (x; k)j = 0. But
sup jf (x; k)j sup jf (x; k) fn(x; k)j + sup jfn(x; k)j
k2I
k2I
k2I
kf fnkBe + sup jfn(x; k)j :
k2I
The rst term can be made arbitrarily small by appropriately choosing n and
the second term vanishes for jxj ! 1.
CHAPTER 2. MATHEMATICAL RESULTS
76
Proof [of part (ii)]. Let f 2 Be . Then for any xed k 2 I , f (; k) 2 B
and therefore Ak f (; k) 2 B . First we show that Af (; ) 2 C (IR3 I ):
j(Af )(x; k) (Af )(x0; k0)j = j(Ak f (; k)) (x) (Ak0 f (; k0)) (x0 )j
j(Ak f (; k)) (x) (Ak f (; k)) (x0 )j
+ j((Ak Ak0 )f (; k)) (x0 )j
+ j(Ak0 (f (; k) f (; k0))) (x0)j : (2.70)
Since Ak f (; k) 2 B the rst term can be made arbitrary small by choosing
jx x0 j small enough. The second term becomes small uniformly in x0 by
choosing jk k0 j small enough since
sup3 j((Ak Ak0 )f (; k)) (x0 )j = k(Ak Ak0 )f (; k)kB kAk Ak0 kL(B) kf kBe
0
x 2IR
and Ak depends continuously on k. The third term in (2.70) yields
sup3 j(Ak0 (f (; k) f (; k0))) (x0 )j kAk0 kL(B) kf (; k) f (; k0)kB
x0 2IR
c max sup
0
jx j>R
jf (x0; k)
f (x0; k0)j;
sup
jx0jR
jf (x0; k)
!
f (x0; k0)j
;
where we used supk2I kAk kL(B) c < 1. This holds because kAk kL(B) is a
continuous function of k on a compact set. The rst term in max(: : :) can
be made arbitrary small by choosing R large since f 2 Be . The second term
vanishes for jk k0j ! 0 since a continuous function on a compact domain
is uniformly continuous.
We now show that
lim sup(Af )(x; k) = jxlim
sup(Ak f (; k))(x) = 0 :
jxj!1
j!1
k2I
k2I
Suppose that this is wrong, then there exists an > 0 and a sequence
fxn; kngn2IN IR3 I with limn!1 xn = 1, such that j(Akn f (; kn))(xn)j >
8n 2 IN. Since I is compact, fkng contains a convergent subsequence (for
simplicity also denoted by fkng) with limn!1 kn = k 2 I . Now
jAkn f (xn ; kn)j jAkn (f (xn; kn) f (xn; k))j + j(Akn Ak )f (xn; k)j
+jAk f (xn; k)j
where the rst two terms get arbitrary small as n ! 1 as has just been
shown, and the third term gets arbitrary small as n ! 1 since Ak f (; k) 2 B .
Thus we have a contradiction and Af 2 Be follows.
2.5. SOME TECHNICAL LEMMAS
77
The estimate for the norm follows directly from
kAf kBe = sup
j(Ak f (; k))(x)j sup kAk kL(B) kf (; k)kB
3
x2IR ;k2I
sup kAk kL(B) kf kBe
k2I
k2I
Lemma 2.28. Let V satisfy (V)3 and let s > 0 such that hxisV (x)
still satises (V)3. Using the notation from the proof of Theorem 2.4 let for
f 2 Be
1 Z e ijkjjx yj hyisV (y)f (y; k) dy ;
2 hxisjx yj
Z e ijkjjx yj
i
k
l
0
s
e
(T f )(x; k) := 2 jkj
hxis hyi V (y)f (y; k) dy ; and
Z kl
f
(Kf )(x; k) := 0 f (x; (kl0; kl)) dkl0 :
e )(x; k) :=
(Tf
kl
Then
f belong to L(Be ) and kK
fkL(Be) 2l , where
(i) the operators Te , Te0 and K
2l is the length of the interval Il .
(ii) The functions hxi s@kl g(x; k) and hxi s(x; (kl0; kl )) belong to Be .
Proof [of part (i)]. Let f 2 B and dene
Z e ijkjjx yj
1
= 2 hxisjx yj hyisV (y)f (y) dy ;
Z ijkjjx yj
(Tk0sf )(x) = 2i jkklj e hxis hyisV (y)f (y) dy ;
then for fe 2 B we have (Tefe)(x; k) = (Tksfe(; k))(x) and analogously for Te 0.
We shall use Lemma 4.27.(ii)
to prove that Te and Te0 are in L(Be ). We have to
0s
s
show that (Tk )k2I and (Tk )k2I are families of operators in L(B ) continuously
depending on k.
Now hyisV (y) still satises the conditions (V)3. According to Lemma
2.7.(i) and (v) hxisTks satises the conditions of Lemma 4.27.(ii). Multiplication by hxi s is a bounded operation in B and thus also (Tks)k2I satises the
conditions of Lemma 4.27.(ii). Hence Te 2 L(Be ).
(Tksf )(x)
CHAPTER 2. MATHEMATICAL RESULTS
78
Next consider Tk0s. From hyisV (y) 2 L1 and
Z
j(Tk0sf )(x)j hxi s 21 jhyisV (y)j dy kf kB
limjxj!1 j(Tk0sf )(x)j = 0 follows. With jei ei0 j j 0 j for ; 0 2 IR we
estimate
jhxis(Tk0sfZ)(x) hx0is(Tk0sf )(x0)j 21 e ijkjjx yj e ijkjjx0 yj jhyisV (y)jjf (y)j dy
Z
1
2 jkj jjx yj jx0 yjj jhyisV (y)j dy kf kB
Z
1
0
jx x jjkj 2 jhyisV (y)j dy kf kB ;
which proves the continuity
of hxis(Tk0sf )(x) in x and thus
that of (Tk0sf )(x)
0s
0s
itself. Therefore Tk 2 L(B ). It remains to show that Tk is norm continuous
with respect to k:
k(Tk0s Tk0s0)f kB =
1 Z hyisV (y)f (y) kl ijkjjx yj kl0 ijk0jjx yj!
= sup3 2
hxis
jkj e
jk0j e!
x2IR 1 Z hyisV (y)f (y) ijkjjx yj kl kl0 sup3 2
hxis e
jkj jk0j x2IR 1 Z hyisV (y)f (y) kl0 ijkjjx yj
0 jjx yj i
j
k
+ sup3 2
e
s
0j e
h
x
i
j
k
x
2
IR
0 k
k
l
l
c jkj jk0j kf kB
Z
s
+ sup3 21 jhyihxVis(y)j e ijkjjx yj e ijk0jjx yj dy kf kB : (2.71)
x2IR
Since we can achieve c jkklj jkkl0j < 2 for any > 0 by choosing jk k0 j small
it remains to show that also
Z jhyisV (y)j 1
ijkjjx yj e ijk0 jjx yj dy < sup3 2
e
hxis
2
x2IR
0
for jk k0 j small. Since hxisV (x) 2 L1 and e ijkjjx yj e ijk0jjx yj 2 there
exists R1 such that
2.5. SOME TECHNICAL LEMMAS
79
Z jhyisV (y)j 1
ijkjjx yj e ijk0 jjx yj dy sup
e
hxis
jxj>R1 2
Z
1
1
sup hxis 2 2jhyisV (y)j dy < 2 :
jxj>R1
Similarly there is R2 such that
Z jhyisV (y)j 1
ijkjjx yj e ijk0jjx yj dy sup3 2
e
hxis
y>R2
x2IR
Z
1
1
sup3 hxis 2
2jhyisV (y)j dy < 4 ;
x2IR
y>R2
holds. Observing that from jxj < R1 and jyj < R2 jx yj < R1 + R2 follows,
we obtain for the remaining part
Z jhyisV (y)j ijkjjx yj e ijk0jjx yj dy sup 1
e
hxis
jxj<R1 2 jyj<R
2
Z
jjkj jk0jj 21 (R1 + R2) jhyisV (y)j dy C jk k0j < 4
for jk k0j suciently small. Combining these results we get that for any
>0
k(Tk0s Tk0s0 )f kB < kf kB
for0 jk k0 j small enough, which proves the norm continuity of Tk0s. Therefore
Tks meets the requirements of Lemma 4.27.(ii) and we conclude that Te0 2
L(Be ).
f. For f 2 Be the continuity of
Finally consider K
f )(x; k) =
(Kf
Z kl
kl0
f (x; (kl0; kl)) dkl0
in x and k is clear. Furthermore
f )(x; k)j lim 2l sup jf (x; k)j = 0 ;
lim sup j(Kf
jxj!1
jxj!1 k2I
f 2 L(Be ) and
so that K
k2I
80
CHAPTER 2. MATHEMATICAL RESULTS
Z kl
f kBe = sup f (x; (kl0; kl)) dkl0 2l kf kBe :
kKf
x2IR3 ;k2I kl0
Proof [of part (ii)]. Since (x; k) 2 B for all k 6= 0 and I is compact,
hxi s(x; (kl0 ; kl)) 2 Be is obvious. Observing
Z
@kl g(x; k) = 2i jkklj e ijkjjx yjV (y)eiky dy
i Z e ijkjjx yj V (y)y eiky dy ;
(2.72)
l
2 jx yj
hxi s@kl g(x; k) 2 Be can be shown using the same types of estimates as in
the proof of part (i) of this lemma.
Bibliography
[1] Agmon, S.: Spectral properties of Schrodinger operators and scattering
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