Methods of modern mathematical physics volume 3 scattering theory michael reed, barry simon

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METHODS OF
MODERN M ATHEMATICAL PH YSIC S
III: SCA TTERIN G TH EO RY

BARRY SIMON

Department of Mathematíes
Duke University

Departments of Mat hematíes
and Physics
Princeton ¡Jniversity

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MICHAEL REED

ACADEMIC PRESS, INC.
Harcourt Braca Jovanovich, Publithars
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London Sydney Tokyo Toronto


C o p y r i g h t © 1979, by A c a d e m ic P r e s s , In c .
ALL RIGHTS RESERVED.
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ACADEMIC PRESS, INC.

United Kingdom Edition published by
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Library of Congress Cataloging in Publication Data
Reed, Michael.
Methods of modern mathematical physics.
Vol. 3 Scattering Theory.
Includes bibliographical references.
CONTENTS: v. 1. Functional analysis.-v. 2. Fourier
analysis, self-adjointness.-v. 3. Scattering theory.-v. 4.
Analysis of operators.
1. Mathematical physics. 1. Simón, Barry,joint
author. II. Title.
QC20.R37 1972
530.T5
7 5 -182650
ISBN 0 -1 2 -5 8 5 0 0 3 -4 (v. 3)
AMS (MOS) 1970 Subject Classifications: 4 7 -0 2 , 8 1 -0 2

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T o M a rth a and Ja ckie


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Scattering theory is the study of an interacting system on a scale of time
and/or distance which is large com pared to the scale of the interaction
itself. As such^it is the most effective means, sometimes the only means,
to study microscopio nature. To understand the im portance of scattering
theory, consider the variety of ways in which it arises. First, there are
variou»phenom ena in nature (like the blue of the sky) which are the result
of scattering. In order to understand the phenomenon (and to identify it as
the result of scattering) one must understand the underlying dynamics and
its scattering theory. Second, one often wants to use the scattering of waves
or particles whose dynamics one knows to determine the structure and
position of small or inaccessible objects. For example, in x-ray crystallography (which led to the discovery of DNA), tom ography, and the
detection of underwater objects by sonar, the underlying dynamics is well
understood. W hat one would like to construct are correspondences that
link, via the dynamics, the position, shape, and intem al structure of the
object to the scattering data. Ideally, the correspondence should be an

explicit formula which allows one to reconstruct, at least approximately,
the object from the scattering data. A third use of scattering theory is as a
probe of dynamics itself. In elementary particle physics, the underlying
dynamics is not well understood and essentially all the experimental data
are scattering data. The main test of any proposed particle dynamics is
whether one can construct for the dynamics a scattering theory that predicts
the observed experimental data. Scattering theory was not always so central
to physics. Even though the C oulom b cross section could have been
computed by Newton, had he bothered to ask the right question, its
calculation is generally attributed to Rutherford more than two hundred
years later. O f course, Rutherford’s calculation was in connection with the
first experiment in nuclear physics.
Scattering theory is so im portant for atomic, condensed m atter, and high


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energy physics that an enorm ous physics literature has grown up. Unfortunately, the development of the associated m athem atics has been much
slower. This is partially because the m athem atical problems are hard but
also because lack of com m unication often m ade it difficult for mathematicians to appreciate the many beautiful and challenging problems in
scattering theory. The physics literature, on the other hand, is not entirely
satisfactory because of the m any heuristic formulas and ad hoc methods.
Much of the physics literature deais with the “ t im e-independent ” approach to
scatteringtheory because the time-independent approach provides powerful
calculational tools. We feel that to use the time-independent formulas one m ust
understand them in terms of and derive them from the underlying dynamics.
Therefore, in this book we emphasize scattering theory as a time-dependent
phenomenon, in particular, as a com parison between the interacting and
free dynamics. This approach leads to a certain im balance in our presen tation
since we therefore emphasize large times rather than large distances. However,

as the reader will see, there is considerable geometry lurking in the background.
The scattering theories in branches of physics as different as classical
mechanics, continuum mechanics, and quantum mechanics, have in common
the two foundational questions of the existence and completeness of the
wave operators. These two questions are, therefore, our main object of study
in individual systems and are the unifying theme that runs throughout the
book. Because we treat so m any different systems, we do not carry the
analysis much beyond the construction and completeness of the wave
operators, except in two-body quantum scattering, which we develop in
some detail. However, even there, we have not been able to include such
important topics as Regge theory, inverse scattering, and double dispersión
relations.
Since quantum mechanics is a linear theory, it is not surprising that the
heart of the mathematical techniques is the spectral analysis of Hamiltonians.
Bound States (corresponding to point spectra) of the interaction Ham iltonian
do not scatter, while States from the absolutely continuous spectrum do.
The mathematical property that distinguishes these two cases (and that
connects the physical intuition with the m athem atical formulation) is the
decay of the Fourier transform of the corresponding spectral measures.
The case of singular continuous spectrum lies between and the crucial (and
o fien hardest) step in most proofs of asym ptotic completeness is the proof
that the interacting Ham iltonian has no singular continuous spectrum.
Conversely, one of the best ways of showing that a self-adjoint operator
has no singular continuous spectrum is to show that it is the interaction
Hamiltonian of a quantum system with com plete wave operators. This deep


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connection between scattering theory and spectral analysis shows the

artificiality of the división of m aterial into Volumes III and IV. We have,
therefore, preprinted at the end of this volume three sections on the
absence of continuous singular spectrum from Volume IV.
While we were reading the galley proofs for this volume, V. Enss introduced new and beautiful m ethods into the study of quantum -m echanical
scattering. Enss’s paper is not only of interest for what it proves, but also
for the future direction that it suggests. In particular, it seems likely that
the m ethods will provide strong results in the theory of multiparticle
scattering. W ehaveadded a section at the end of this Chapter (Section XI. 17)
to describe Enss’s method in the two-body case. We would like to thank
Professor Enss for his generous attitude, which helped us to include this
material.
The general remarks about notes and problems made in earlier introductions are applicable here with one addition: the bulk of the material
presented íh this volume is from advanced research literature, so m any of
the problems are quite substantial. Some of the starred problems summarize
the contents of research papers!


Contents

XI:

SCATTERIN G TH EO RY

1. An overview o f scattering phenomena
2. Classical partióle scattering
3. The basic principies o f scattering in Hilbert space
Appendix 1 Stationary phase methods
Appendix 2 Trace ideal properties o ff(x )g ( — iV)
Appendix 3 A general invariance principie for wave operators
4. Quantum scattering I: Two-body case

5. Quantum scattering II: N-body case
6. Quantum scattering III: Eigenfunction expansions
Appendix Introduction to eigenfunction expansions by the
auxiliar y space method
7. Quantum scattering IV: Dispersión relations
8. Quantum scattering V: Central potentials
A. Reduction o f the S-matrix by symmetries
B. The partial wave expansión and its convergence
C. Phase shifts and their connection to the Schródinger
equation
D. The variable phase equation
E. Jost functions and Levinsons theorem
F. Analyticity o f the partial wave amplitude for generalized
Yukawa potentials
G. The Kohn variational principie

ini
ix
xv

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Preface
¡ntroduction
Contents o f Other Volumes

1
5
16
37

47
49
54
75
96

112
116
121
121
127
129
133
136
143
147


Appendix 1 Legendre polynomials and spherical Bessel
functions
Jost Solutions fo r oscillatory potentials
Jost Solutions and the fundamental problems o f

149
155

scattering theory
9. Long-range potentials
10. Optical and acoustical scattering I: Schródinger operator
methods

Appendix Trace class properties o f Greerís functions
11. Optical and acoustical scattering II: The Lax-Phillips
method
Appendix The twisting trick
12. The linear Boltzmann equation
13. Nonlinear wave equations
Appendix Conserved currents
14. Spin wave scattering
15. Quantum field scattering I: The external field
16. Quantum field scattering II: The Haag-Ruelle theory
17. Phase space analysis o f scattering and spectral theory
Appendix The RAG E theorem
Notes
Notes on scattering theory on C*-algebras
Problems

164
169

MATERIAL PREPRINTED FROM VO LU M E IV

X III.6
X I I 1.7

X III.8

The absence o f singular continuous spectrum I: General
theory
The absence o f singular continuous spectrum II : Smooth
perturbations

A. Weakly coupled quantum systems
B. Positive commutators and repulsive potentials
C. Local smoothness and wave operators fo r repulsive
potentials
The absence o f singular continuous spectrum I I I:
Weighted l i spaces
Notes
Problems

List o f Symbols
Index

184
203

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Appendix 2
Appendix 3

210
241
243
252
278
285
293
317
331
340

344
382
385

406
411
421
427
433

438
447
450
455
457


Contents of Other Volumes

Volume I:

Preliminaries
Hilbert Spaces
Banctoh Spaces
Topologicai Spaces
Locally Convex Spaces
Bourtded Operators
The Spectral Theorem
Unbounded Operators

Volume II:

IX
X

Fourier Analysis, Self-Adjointness

The Fourier Transform
Self-Adjointness and the Existence o f Dynamics

Volume IV :

X II
X III

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/
II
II I
IV
V
VI
V II
V III

Functional Analysis

Analysis of Operators

Perturbation o f Point Spectra
Spectral Analysis

Contents of Future Volum es:

Convex Sets and Functions, Conunutative
Banach Algebras, Introduction to Group Representations, Operator Algebras,
Applications o f Operator Algebras to Quantum Field Theory and Statistical
Mechantes, Probabilistic Methods

XV


XI:

Scattering Theory

XI.1

An overview of scattering phenomena

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¡t is notoriously difficult to ohtain reliable results for quantum mechanical scattering problems.
Since they involve complicated interference phenomena o f waves%any simple uncontrolled approximation is not worth more than the weather forecast. However, for two body problems with
central forces the Computer can he used to calcúlate the phase shifts . . . .
W. Thirring

In this chaptcr we shall discuss scattering in a variety of physical situations. O u r main goal is to illustrate the underlying similarities between the
large time behavior of many kinds of dynamical systems. We study the case

of nonrelativistic quantum scattering in great detail. O ther systems we treat
to a Iesser extent, emphasizing simple examples.
Scattering normally involves a com parison of two different dynamics for
the same system: the given dynamics and a “ free” dynamics. It is hard to
give a precise definition of “ free dynamics ” which will cover all the cases we
consider, although we shall give explicit definitions in each individual case.
The characteristics that these free dynamical systems have in com m on are
that they are simpler than the given dynamics and generally they conserve
the m om entum of the “ individual constituents” of the physical system. It is
im portant to bear in mind that scattering involves more than just the interacting dynamics since certain features of the results will seem strange otherwise. Because two dynamics are involved, scattering theory can be viewed as
a branch o f perturbation theory. In the quantum-m echanical case we shall
see that the perturbation theory of the absolutely continuous spectrum is


2

XI:

SCATTERING THEORY

lim ( 7 ; p - TJ°V _) = 0
t — - 00

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involved rather than the perturbation theory of the discrete spectrum
discussed in C hapter XII.
Scattering as a perturbative phenom enon emphasizes tem poral asymptotics, and this is the approach we shall generally follow. But all the concrete
examples we discuss will also have a geometric structure present and there is
clearly Iurking in the background a theory of scattering as corretations

between spatial and tem poral asymptotics. This is an approach we shall not
explicitly develop, in part because it has been discussed to a much lesser
degree. We do note th at all the “ free” dynamics we discuss have “ straightline motion ” in the sense that Solutions of the free equations which are
concentrated as t -» — oo in some neighborhood of the direction n are concentrated as f -* + oo in a neighborhood of the direction —n. These geomet­
ric ideas are useful for understanding the choice of free dynamics in Sections
14 and 16 where a piece of the interacting dynam ics generates the free
dynamics. And clearly, the geometric ideas are brought to the fore in the
Lax-Phillips theory (Section 11) and in Enss’s m ethod (Section 17).
Scattering theory involves study ing certain States of an interacting system,
namely those States that appear to be “ asym ptotically free” in the distant
past and/or the distant future. T o be explidt, suppose that we can view the
dynamics as transform ations acting on the States. Let T, and T}0> stand for
the interacting and free dynam ical transform ations on the “ set of States ” E.
I may be points in a phase space (classical mechanics), vectors in a Hilbert
space (quantum mechanics), o r Cauchy d ata for some partial differential
equation (acoustics, optics). O ne is interested in pairs < p _ , p> e E so that

for some appropríate sense of limit, and similarly for pairs that approach
each other as t -* + oo. O ne requirement that one m ust make on the notion
of limit is that for each p there should be at m ost one p . .
The basic questions of scattering theory are the following:

(1)
Existence o f scattering States Physically, one prepares the interact­
ing system in such a way that some of the constituents are so far from one
another that the interaction between them is negligible. O ne then “ lets go,”
that is, allows the interacting dynamics to act for a long time and then looks
at what has happened. O ne usually describes the initial State in term s of the
variables natural to describe free States, often m om enta. O ne expects that
any free State “ can be prepared,” that is, for any p _ e E, there is a p e E with

lim ,.,.*, T , p - T{0>/)- = 0. Proving this is the basic existence question of
scattering


(2) Uniqueness o f scattering States In order to describe the prepared
state in terms of free States, one m ust know that each free State is associated
with a unique interacting State; th at is, given p . there is at m ost one p such
that T¡0)p - - T ,p -> 0 as í-> - o o . Notice that this is distinct from the requirem ent on the limit above that there should be at most one p . for each p.

E,„ = p e E |3 p _ e E w i t h
!

lim T<0)p . - T , p = 0
t-* ~ ao

and
¡ p e E | 3 p + e E w ith lim T ™ p+ - T , p = 0
I

r-> + oo

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(3) Weak asymptotic completeness Suppose that one has an interacting
State p th at looked like a free state in the distant past in the sense that
lim ,.,-*, T \0)p - — T,p = 0 for some state p_ .O n e hopes that for largepositive times, the interacting state will again look like a free state in the sense
that there exists a state p+ so th at lim,_ + 00 T¡0,p+ — T,p = 0. In order to
prove this, one needs to show that the two subsets of E

are ftqual. If in fact Ein = E w, , then the System is said to have weak asymp­

totic completeness.

(4) Definition o f the S-transformation If one has a pair of dynamical
Systems scattering States (both as t -* —oo and as í -* oo) and for which weak asymp­
totic completeness holds, then one can define a natural bijection of E onto
itself. Given p e E, existence and uniqueness of scattering States assures us
that there exists a state Q *p e E,n with lim,_ _ *,
p) — T{0)p) = 0. Similarly, Í2~ is defined by lim,_ + a0 (T,(Q~p) — T \O)p) — 0. Cl* (respectively,
Q " ) is a bijection from E onto E in (respectively, E^,,). W eak asymptotic
completeness assures us that E¡„ = E ou), so one can define the bijection
S = ( Í T )-* Q + : E - E

S is called the scattering transformation. Thus, T \0)(Sp) and T \0)p are related
by the condition that there exists a state
= Cl*p = Cl~(Sp)) so that T,{¡/
“ interpolates” between them. T hat is, T,\¡/ looks like T \0)p in the past and
T}0)Sp in the future. Thus S correlates the past and future asymptotics of
interacting histories. The reader should be wamed that the maps
S' = £2+(í2- )~ ‘ : Ein-^Eom and also the maps ( íl+)-1 íl~ and í l ” ( íl+)_1
occasionally appear in the literatura. When weak asym ptotic completeness
holds, S' = £1" S(Q~)" *, so S and S' are “ similar.” F or this reason, the choice
between S and S ’ is to some extent a m atter of personal preference. We use S,


4

XI:

SCATTERING THEORY

the so-called EBFM S-matrix, throughout this book. W ediscuss the reasons
for the ± convention in Sections 3 and 6.
In classical partióle mechanics S is a bijection on phase space. In a quan­
tum theory with weak asym ptotic completeness S is a linear unitary transformation and is called the S-operator or occasionally the S-matrix.

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(5) Reduction o f S due to symmetñes In many problems there is an
underlying symmetry of both the free and interacting dynamics. This allows
one to conclude a priori, without detailed dynamical calculations, that S has
a special form. See Sections 2 and 8 for explicit details.
(6) Analyticity and the S-transformation A common refinement of scat­
tering theory for wave phenom ena (quantum theory, optics, acoustics) is the
realization of S or the kernel of some associated integral operator as the
boundary valué of an analytic function. In a heuristic sense this analyticity is
connected with Theorem IX. 16. F or schematically, S describes the response
R of a system to some input / in the following form:

- ao

This formula has two features built in: (i) time translation invariance, that is,
/ is a function of only t — t'; (ii) causality: R (t) depends only on Z(í') for
t’ <>t. Thus / is a function on [0, oo). Its Fourier transform is thus the
boundary valué of an analytic function. It is this causality argument that is
intuitively in the back of physicists’ minds when discussing analytic properties. Unfortunately, the proofs of these properties do not go along such
simple lines. We shall restrict our detailed discussion of analyticity to the
two-body quantum-mechanical case (Section 7) and to the Lax-Phillips
theory (Section 11).

(7)
Asymptotic completeness Consider a system with forces between its
components that fall off as the com ponents are moved apart. Physically, one
expects a state of such a system t o 4i decay ” into freely moving clusters or to
remain “ bound.” In many situations, there is a natural set of bound
States, IboUnd c I . O ne can usually prove that Sbound n £¡n = 0 . The above
physical expectation is
(0
44 + ” is difTerent in classical and quantum -m echanical systems. In classical
particle mechanics “ + ” indicates set theoretic unión; in quantum theory it
indicates a direct sum of Hilbert spaces. Establishing that (1) holds is the
problem of proving asymptotic completeness. Notice that asymptotic com­
pleteness implies weak asym ptotic completeness. We remark that implicit in


the idea that each free State has an associated interacting State is the assumption that the free dynamics has no “ bound ” States.

XI.2

Classical p artid a scattering

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We emphasize that the above description is schematic. In each physical
theory there are complications, and various modifícations must be made.
Among these are: (i) In classical mechanics Z comes equipped with sets of
measure zero and the natural interpretation of statements like Z ¡n =
is
that they diflfer by sets of measure zero. (ii) In some systems, including
m any-body systems, the State spaces of the free and interacting dynamics are

different (see Sections 5,15, and 16). (iii) In quantum -mechanical systems one
can define an S-operator even without weak asymptotic completeness (see
Section 4). Weak asymptotic completeness then becomes equivalent to the
unitarity of S. (iv) In certain very special cases the free dynamics may have
bound States (see Section 10). (v) In the Lax-Phillips theory (Section 11) the
free dynamics is replaced by the geometric notion of “ incom ing” and
“ outgoing” subspaces.
Usually, the«iinteracting dynamics is obtained initially by perturbing a
simple dynamics which then plays the role of the “ free ” dynamics. However,
in some special physical theories there is no natural unperturbed dynamics
to coirfpare with the interacting dynamics. In such cases one first isolates
certain especially simple Solutions of the interacting system. Then one tries
to describe the asymptotic behavior of the complete interacting system in
terms of the interactions of these simple Solutions. M agnon scattering (Sec­
tion 14) and the Haag-Ruelle theory (Section 16) are examples of such
systems, as is the scattering theory for the Korteweg-deVries equation,
which we do not treat.

The simplest system with which to illustrate the ideas of scattering theory
is the classical mechanics of a single particle moving in an extem al forcé field
F(r). This theory is equivalent to the scattering of two particles interacting
with each other through a forcé field F (rj — r 2) because the center of mass
m otion of such a two-body system separates from the motion of
r i 2 = r i ” r2 • We shall suppose that the particle has mass one, which is no
loss of generality.
The States of such a single particle system are points in phase space, that is,
a pair (r, v) e R6 representing the position and velocity of the particle. The
free dynamical transform ation is given by T¡0)

6

XI:

SCATTERING THEORY

free dynamics conserves the velocity. The interacting dynam ics is given by
7j(r0 , v0) = (r(t), v(i)) where v(r) = f(f) and r(í) solves the equation
i{t) = F(r(t))

(2a)

with initial conditions
r(0) = r0 ,

r(0) = v0

(2b)

In order to be sure th at (2) has a unique solution for all times, we shali
suppose that

|F ( r ) - F ( r ') | < ; D * | r - r '|

for all r
if

|r - r '|^ l

(3a)

and

|r | < R

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| F (r) |

(3b)

where DR is an £-dependent constant. The techniques we developed in
Section V.6 assure us th at (2) has a unique solution for small time if (3b)
holds, and it is not hard to prove th at this solution exists for all times (see
Proposition 1 in the appendix to Section X.1 and Problem 1). The only place
where the conditions (3) enter in the theory that we shall develop is in
establishing this global existence and uniqueness. If one can establish this by
some other means, (3) can be dispensed with and conditions (4) below need
be required to hold only for large distances. In particular, local repulsive
singularities present no problem.
To establish the existence and uniqueness of scattering Solutions, we shall
need to have further restrictions on the forces. These restrictions, which
require that the interaction between constituents falls off as r-> oo, where
r «■ | r | , are typical of scattering theories. Speciñcally, we shall suppose that:
| F (r) | < ,C r~a
| F(x) - F (y)| < D r~f |x — y |

for all r and some a > 2
for all x, y with
x, y ^ r and some 0 > 2

(4a)

(4b)

Under these assum ptions we shall prove the existence and uniqueness of
scattering Solutions. O ne can establish existence using only (4a) (Problem 2),
but uniqueness requires the Lipschitz condition (4b) (Problem 3). This is
reminiscent of the situation we encountered in Section V.6 when discussing
Solutions of differential equations with initial conditions. Lipschitz condi­
tions were also required there for uniqueness. This is not surprising since
according to our intuitive picture in Section 1, scattering Solutions can be
viewed as Solutions obeying “ initial conditions at t ■» —oo.”
The conditions (4) do not include the im portant case of Coulom b scatter­
ing where the theory m ust be modified. We discuss this case in Section 9.


Henceforth we shall d rop the boldface notation for vectors except in the
statem ents of theorems and in situations where confusión might arise be­
tween a vector and its length.
T h e o r e m X I.1 (existence and uniqueness of scattering Solutions; classical
particles) Let F(r) be a function from R3 to R3 obeying (3) and (4). Let
(r
R6 be given with v.*,
Then there exists a unique solution of (2a) obeying
(5a)

lim |r(r) — r _ ^ — v _ 00f | = 0

(5b)

and
t-* - 00

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lim |r(í) — v . ^ l = 0
t-* - 00

Proof Sincewe are assuming (3), by the above remarks it is sufficient to
prove the existence and uniqueness of Solutions in ( —oo, T) for some T. In
keeping with the idea that scattering Solutions obey initial conditions at
t = —oo, it is natural to use the m ethod of Section V.6.A and rewrite the
differential equation as an integral equation. In fact, one can show (Problem
4) that* r(r) obeys (2a) and (5) on ( —oo, T) if and only if r(f) =
+
v - a t + n(í), where u is continuous and satisfies
M(0 =

[

f
- ao

F (r -*> + v . x x + u(x)) dx ds
- 00

where the integral converges absolutely.
Choose T < 0 so that:
(i)
(ii)

(iii)
(iv)

I r . . + »_«,C(« - l ) - ‘(« - 2)-> |J p _ . |- * |T |2- < 1;
y m D W - I ) ' 1V - 2 ) - 1
I- ' 1 7 f - ' < 1;
¿ | T | !» _ „ ! > 1.

( 6)

Here C, a, D, fi are the constants in condition (4). Now suppose that u(t) is
an R3-valued continuous function on ( —oo, T) with || u|| ^ < 1. Let r(t) =
+ v - a t + u(í). (i) and (iv) assure us that |r(r)| ^ ¿ | t |
|. By (4a),
the integral
J.* , ^ ( r . * , +
x + m(t))| dx ds converges absolutely.
Let
J ( T = {« e C( —oo, T ) with valúes in R3 ¡ HuH* ^ 1}
and define

J t T -* J t T by

(^ H )(t) = J |

~ 00 ~ 00

+ v-^x + m (t)) dxds

8

XI:

SCATTERING THEORY

(4a) and (ii) assure us that ||& u\\m < 1 if Hm^ < 1, so 3F maps the complete
metric space J ( T into itself. (4b) and (iii) imply that
\\F u - PvWn < y\\u - vWn

We now define two im portant maps:
D e fin itio n Let Z = R6 and let
totic to a + bt at - o o . Set Z0 =
í l +: I 0 Z is defined by

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so 9 is a contraction on M T since T was chosen to make y < 1. Thus, by the
contraction m apping principie (Theorem V.8), & has a unique fixed point in
J Í T. It is easy to prove now that (6) has a unique solution. F o r if u x and u2
both solve (6), then both will lie in J t T for some T < T. But by the above
argument, there is a unique solution of (6) in J t T for any T < T, so u 1 = u2
on (~oo, T'). By the uniqueness of Solutions with initial conditions at
- T — 1, M| = u2 on ( —oo, T). |

be the solution of (2a) asymp^>1^ = 0}- Then the wave operator

Í T < a, b ) = < ^ « > ( 0 ) ,

0 )>

Similarly, £1" is defined by
Q "< a, b> - <rft¡»>(0), r^ > (0 )>

Thus Q +w is that point of phase space which is the t = 0 initial data for a
solution of the interacting equations of m otion which is asymptotic at
t = - oo to the solution of the free equations of m otion with data at t = 0
equal to w.
The wave operators have several im portant properties:

Theorem X I.2
Suppose that conditions (3) and (4) hold for a forcé field
F(r) and let Q* be the associated wave operators. Then:

(a)

Let Tt and T¡0) be the interacting and free dynamics, respectively. Then
for all w € Z0 ,
n*w -

lim 71, T \0)w
t—* ao

where the limits are uniform on compact subsets of £ 0 •
(b) fi* T[0) = T, Q* on Z0 for all s.


(c)
(d)

(e)

(isometry of Q * ) If F is conservative, that is, if F * —VK for some
function V, then Q* are measure-preserving transformations.
If F is conservative and K (r)-+ 0 as r-» o o , then E (íl±w) = £ 0(w)
where £(r, v) = \ v 2 + K(r) and £ 0(r, v) = |t>2.
If F is C°° and
3,a|F(r)
dr\' - d i ?

^ n r -w -2 -«

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for all r, a and some e > 0, then Q* are C°° maps.
Proof (a) This is a typical property of íí* and wtll be used to define the
analogues of íl* in quantum -m echanical situations. Since Q +x = y means
that lim ,.,-*, \ T,y — r j 0,x | = 0 and (7¡)- ‘ = 71,, (a) is intuitively expected. We shall prove the formula for Ó +; the proof is essentially identical
for Q “ . F or fixed I e R , define J t T as before. F or (a , b ) e L 0 , t <> T, and
u e J t T , defipe the function
r u on ( - oo, T ) by
(^a!b. t u)(s) = f í F(a + bz + u(t)) dz da
Jt

Let
« be of the same form with t = - oo. O ne now proves the following three facts (Problems 5, 6):
(i)

F o r any compact K c: E0 , we can find T < 0 so that for (a , b ) e K
and t e ( —oo, T),

T takes J t T into itself and is a contraction. The
constant y in the equatioil \\^ ! b . r « rằ IL ^ VĂô - ằIIđ may
be chosen, independently of (a , b ) e K and f e ( - o o , T), to be less
than 1.
(ii) If K and
T are as defined in
(i), for any u e J t T,
lim, ^ _ „ \ T u =
u- The convergence is uniform on J t T
and K.
(iii) A general result about contractions: Suppose that F„ form a family of
m aps of a complete m etric space to itself. If p(Fn p, Fmq) <>cp(p, q) for
all p, q, n and some c < 1, if l i m ^ ^ Fnp = Fx p for all p, and if pH
(respectively, pTO) are the unique fixed points of F„ (respectively, £ „ ),
then lim ,^ ^ pn = p x . M oreover, the rate at which p„ converges to p»
depends only on the rate at which F„pOD converges to £ „ p^ = p^
and c.
Let <>,. T be
lim ,...* uí,',V r =

the

fixed

point

of

t

■ We

conclude

that

N ow , using the fact that T_r+1 is continuous from


E to I , we conclude the proof of (a):
Q +<«, b} - T_r + , <a + b(T - 1) + ul;¿°'T(T - 1),¿> + ««X M T - 1)>
= Iim T_r+ jt~* - ao

*

*

lim 7 L
t-*- 00

1 T _ f ^ r _ | T\°\a> by

lim T ^ \ a y by

(b) This is a general consequence of (a) since
0*71°^=

lim 7Lf 7<0¿ w «
t-* T ao

lim 71t+a7<0)w - T.f}**
x-+ £ ao

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t- * ~ a o

Here we have used the continuity of 7¡ and the fact that as t -► ± oo, t = s 4r -» ± oo for fixed s.
(c) This is another general feature of scattering theory which we shall
meet in quantum scattering in a slightly different guise. F o r conservative
systems, it is known that Tt is measure-preserving (Theorem X.78). Similarly,
T}0> is measure-preserving, so T_, T}0) is measure-preserving for all t. Let /
be a continuous function of com pact support on E0 • Then, by (a),
f / ( f l +w) d 6w = lim | f ( T . , T J0,w) d 6w = | f{ w ) d 6w
t~* ~ ao

Thus í l +, and similarly Í2", are m easure-preserving maps.
(d) Follows from (a), the conservation of energy (E » T, = E) and the
assumption that V -* 0 as r -» oo.
(e) Under the hypothesis,
u is a C® m ap of E0 x * ^ r into J í T
(Problem 7). By a general theorem on sm oothness of fixed points of contractions (Problem 5b) the fixed points of
and henee their valúes at
r = T — 1 are C®. Since T, is a C® m apping for each r, propagating the
solution from t = T — 1 to t = 0, we conclude that Q* are C® maps. |
The domains of Q* are all of £ m inus a set of measure 0. In general, the
range of £1* will not be all of £ or even E minus a set of measure zero.

Example Let F obey the hypotheses of (d) of Theorem XI.2. Then

Ran £í+ £ « V , b'y | | | h ' | 2 + V{á) > 0}. The set

has nonzero measure if V is continuous and negative at any point.


D e f in itio n
Let £¡„ = Ran Q +,
= Ran Cl , and lct L bMjnd be the set
of (r, v) so that the solution r(t) of (2) satisfies
sup | r(f) | + sup | r(f) | < oo
r
r

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Thus, bound States are those whose trajectories lie in bounded regions of
phase space. W eak asym ptotic completeness says that Z in = £<*,,, and
asym ptotic completeness that Z¡„ = Z ou, = £ \ £ bound. Since we have already
thrown out sets of m easure zero (namely, {(a, í>) | b = 0}) in defining Q *, we
should be prepared to have these equalities m odulo sets of measure zero. In
general, there do exist Solutions that are asymptotically free as t -» - oo but
not as í -» 4- oo (capture; see Problem 9).
If the forcé is conservative, that is, F(r) = - VK(r), then by our hypotheses
on F, V is sm ooth and bounded. In this case, by conservaron of energy,
|r( í) | is autom atically bounded, so <r, t>> e Zbcmd if and only if
su p ,|r(f)| < o o .

Theorem XI.3 (asymptotic completeness; two-body classical particle scat­
tering)
Let F (r) = —VK(r) with F - » 0 as r-* o o . Suppose also that F

obeys (3) and (4). Then £ in,
, and £ \E boumi agree up to sets of measure 0.
Proof Let

„(í) be the solution of r(í) = F(r(f)), r(0) = q, r(0) = v. Define
**

o)

►
± 00

We first want to show that N + and JV_ agree up to sets of measure 0, that is,
/i(jV+yV_) + p(N_\JV+) = 0 w herep is Lebesgue measure. The measurability of sets like N + , N - , £(*,„„subsets of R6 with [ j K ñ = R6, K„ c K¡,nl ,. Let N {? = {(q, t>> | T,(q, r> 6 K„
for all t 6 [0, oo)} and similarly for N {*\ W e first note that N ± = (j„
for,
using conservation of energy, if lim,_ + „ | rq v(t) | < oo, then 7jcom pact subset of R6 as t runs from 0 to oo. Thus, if p e N+ \ N _ ,
p e N (J)\N <") for some n. Therefore, it is sufficient to prove that
P ÍN ^A N Í* ) = 0 for each n. Let T, be the interacting dynamics. We first note
that p |¿°-1 Tk
<= N*"* and that N l? => T,
=> T2 N (? =>•••. Thus
p(jv?\Atf?) <

f \ T kN ^ < Í m n í U n í ’)

But, by Liouville’s theorem, fi(Tk N i? ) = pi(N{+) < oo. Since Tk N*? c
we conclude that fi(N ll )\Tk N {1)) - 0, so p(lV«!?\iV«?)" 0. A similar proof

shows that p(N _ \N + ) = 0 so fi(N + A N - ) = 0.


Now suppose r(t) solves N ewton’s equation and
|r(í)| = oo. We
shall first show that if the energy E(r(0), r(0)) > 0, then |r( t) | > C |í | for t
large and use this to prove that r(t) approaches a free solution. Let l(t) =
$ |r(r)|2 be the moment of inertia. Then
¡(t) = f • r = r(f)r(í)
where r(í)= |r(t)| and r(í) = dr/dt (which is not equal to | dr/dt | in
general). Also,

= 2E + r • F(r) - 2K(r)

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r ( t ) = r ( t r + F (r(t))-r(t)

Since E > 0, and both r • F and V go to zero as r —> oo, we can find R 0 so
that |r | > R 0 implies |r • F(r) — 2V(r)\ < E. Since Iim ,..* |r(í)| = oo, we
can find some í0 with r(t0) > R 0 , r(t0) > 0. We now claim that r(t) > R0 for
all t > t0; for if not, let í , be the smallest t > t0 with r(f) = R 0 . Then Y(t) > E
for r e [ f 0 , f , ] so that /( t ,) = r ( í,) r ( t,) > /(í0) > 0. Since r ( t ) > R 0 for
t=
and r ( f ,) = R 0 , we know that r(ít ) < 0, and thus we have a
contradiction. It follows that r(í) > R0 for all t > t0 and therefore for all
f > t 0 , I(t) > a + bt + E t2/2 for suitable constants a and b. Thus
r(t) > \ty jE for t sufficiently large. Using (4), we know that
F(r(t)) dt
exists, so we can define

and
a = r(f0) — bt0 — f

f

F(r(t)) dt ds = lim(r(f) — bt)

The second integral also exists. Moreover,
lim | r(í) — a — bt | + | r(t) — £>¡ = 0

Thus, if E > 0 and l i m , ^ | r(t) | = oo, then r(í) is a scattering solution, that
is, .
Now, let L' be L with two sets of measure zero removed: namely,
N* A N ~, which has measure zero by the first part of our proof; and
{(r, p) | E(r, p) = 0}, which has measure zero since {p ¡ E(r0 , p) = 0} is a
sphere that has measure zero for each fixed r0 . Suppose that w e Z '\Z bound
and let r(t) be the solution of (2 )jw th <r(0), r(0)> = w. Since w i Z bound,
either lim ,^ -* |r(t)| = oo or lim,^ +00 |r( í) | = oo so w e (Z\N + ) u
(Z\N~). Since w i N + A N~ - (Z\N +) A (Z\AT), we must have


w e (L \N * ) n (£\N ). By the second part of our argument, since E{w) ± 0,
we have w e I¡„ and w e I oul. This proves that I'\^bound = í ' n
=

2 ' n S in. |
Now that we
S-transformation.

have

asym ptotic

completeness

we

define

the

sw = ( n - ) - ‘( n +w)
Thus one has the picture shown schematically in Figure XI. 1.

F ig u r e

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Definition Let I (±) = ( íí* )- ‘[E'^bound]- The S-transformation is the
m ap S : X(+) - » Z<_) defined by

XI. 1 Schematic picture of scattering.

The S-transform ation has thus been defined as a m ap from R6 to R6, or
rather from R6 minus a set of measure 0 to R6. As a final topic in classical
scattering theory, we shall describe a way of “ reducing S ” to two real-valued
functions of two real variables in the case that F is a central forcé, that is,
K(r) is a function of |r| = r alone. First we note some symmetries of the

S-operator. Since Q ± T it0 )= 7;Q *, ST¡0) = T¡0)S. Since E(Q±w) E0(w),
E0(Sw) = E0(w). Finally, rotational invariance of F has two consequences.
Let R be an element of SO(3), the family of rotations on three-space. Define
R on I by K<r, t>> = M oreover, the angular momentum L<r, v> = v x r is conserved, so L(Sw) =
L(w). We summarize:
P ro p o sitio n

(a)
(b)
(c)
(d)

S T \0) = T \0)S.
SR = RS.
E0(S •)=£<>(•).
L(S •) = L( •).


F i g u r e X I.2

Central scattering.

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Conditions (a) and (b) allow us to reduce S to a vector-valued function of
only two variables. F or the family of sets {R7'J0)w |f 6 R, R e S0(3)} foliates
X into a two-parameter family of four-dimensional manifolds (with some
exceptional manifolds of sm aller dimensión), the manifolds of constant £ 0
and | L | . By (a) and (b) if we know Sw for one w from each such manifold,

we know 5 for all w. Because of (c) and (d), Sw can lie only on a twodimensional manifold where £ 0 and L are equal to their valúes at the point
w. Thus we expect 5 to be param etrized by two real-valued functions of two
real variables.
Let us be more explicit: By rotational invariance of S, it is enough to know
S(r, v) when v = pz and when r is in the y, z plañe, where z is a unit vector in
the z direction. If S (r, v ) = (r ', v'), then by property (a), S ( r + vr, v) =
( r ' + v'í, v'), so we may suppose that r • v =» 0 or r = by. T o summarize, S
may be recovered if we know S (b y, p z) for all real num bers b and p. Let
S(by, pz) = (r', v'). By conservation of energy | v' | = p so v' = pe(h, p)
where é is some unit vector. By conservation of angular mom entum, r' and v'
lie in the y, z plañe and the com ponent of r' perpendicular to v' is
determined. There are thus two functions that describe S: the scattering
angle 6 = arccos(¿ • z) and the time d eb y T — r' • e/p. These are written as
functions of the m om entum p and impact param eter b, or equivalently as
functions of the energy E = \ p 2 and angular m om entum t — pb. O ne thus
has the picture shown in Figure XI.2. Actually, one can explicitly solve the

central two-body problem up to quadratures and prove (see Problem 11 or
the reference in the Notes):

(7a)
W . E)
r = 2 f°°{[2E - r " V 2] -1/2 - [2£ - 2K - r~ 2¿ 2] - 112} i r

(7b)