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

In order to understand the phenom enon and to identify it as the result of scattering one must understand the underlying dynamics and its scattering theory.. Therefore, in this book we e

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METHODS OF

MODERN M ATHEM ATICAL PH Y SIC S

III: SCA TTER IN G TH EO RY

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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 th eo ry -v 4

Analysis of operators.

1 Mathematical physics 1 Simón, B arry,joint

author II Title.

T o M a rth a and J a c kie

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 T o 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 phenom enon (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 determ ine the structure and position of small or inaccessible objects F or example, in x-ray crys- tallography (which led to the discovery of DNA), tom ography, and the detection of underw ater 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 approxim ately, 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 experim ental 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 experim ental data Scattering theory was not always so central

to physics Even though the C oulom b cross section could have been com puted by Newton, had he bothered to ask the right question, its calculation is generally attributed to Rutherford more than tw o hundred years later O f course, R utherford’s calculation was in connection with the first experim ent in nuclear physics

Scattering theory is so im portant for atomic, condensed m atter, and high

energy physics th at an enorm ous physics literature has grown up U n- fortunately, the developm ent of the associated m athem atics has been much slower This is partially because the m athem atical problem s are hard but also because lack of com m unication often m ade it difficult for m athe- maticians to appreciate the m any 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 m ethods 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 tim e-independent formulas one m ust understand them in term s 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 ou r presen tation since we therefore em phasize large times rather than large distances However,

as the reader will see, there is considerable geom etry lurking in the back- ground

The scattering theories in branches of physics as different as classical mechanics, continuum mechanics, and quantum mechanics, have in com m on the two foundational questions of the existence and completeness of the wave operators These two questions are, therefore, ou r m ain object of study

in individual systems and are the unifying them e th at 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 m athem atical techniques is the spectral analysis of Ham iltonians Bound States (corresponding to point spectra) of the interaction H am 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 form ulation) 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 m ost proofs of asym ptotic completeness is the proof that the interacting H am iltonian has no singular continuous spectrum Conversely, one of the best ways o f showing th at a self-adjoint operator has no singular continuous spectrum is to show th at it is the interaction Hamiltonian of a quantum system with com plete wave operators This deep

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.

W hile we were reading the galley proofs for this volume, V Enss intro- duced 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 m ultiparticle scattering W ehaveadded a section at the end of this C hapter (Section XI 17)

to describe Enss’s m ethod 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 rem arks about notes and problems m ade in earlier intro- ductions are applicable here with one addition: the bulk of the m aterial presented íh this volume is from advanced research literature, so m any of the problem s are quite substantial Some of the starred problem s summarize the contents of research papers!

3 T he basic principies o f scattering in Hilbert space 16

Appendix 3 A general invariance principie fo r wave operators 49

Appendix Introduction to eigenfunction expansions by the

B The partial wave expansión and its convergence 127

C Phase shifts and their connection to the Schródinger

F Analyticity o f the partial wave amplitude fo r generalized

Appendix 1 Legendre polynomials and spherical Bessel

11 Optical and acoustical scattering II : The Lax-Phillips

16 Quantum field scattering II: The H aag-Ruelle theory 317

17 Phase space analysis o f scattering and spectral theory 331

MATERIAL PREPRINTED FROM V O LU M E IV

X I I I 6 The absence o f singular continuous spectrum I: General

X I I 1.7 The absence o f singular continuous spectrum I I : Smooth

C Local smoothness and wave operators fo r repulsive

Contents of Other Volumes

I X The Fourier Transform

X Self-Adjointness and the Existence o f Dynamics

X I I Perturbation o f Point Spectra

X I I I Spectral Analysis

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

X V

XI: Scattering Theory

¡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 approxi-

mation 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

XI.1 An overview of scattering phenomena

In this chaptcr we shall discuss scattering in a variety of physical situa- tions O u r main goal is to illustrate the underlying similarities between the large time behavior of m any kinds of dynam ical systems We study the case

of nonrelativistic quantum scattering in great detail O ther systems we treat

to a Iesser extent, em phasizing simple examples

Scattering norm ally involves a com parison of two different dynam ics for the sam e 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 dynam ical systems have in com m on are that they are sim pler 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 m ore than just the inter- acting dynamics since certain features of the results will seem strange other- wise 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

involved rather than the perturbation theory of the discrete spectrum discussed in C hapter XII

Scattering as a perturbative phenom enon emphasizes tem poral asym ptot-

ics, 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 o f 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 m uch lesser degree We do note th at all the “ free” dynamics we discuss have “ straight- line motion ” in the sense th at Solutions of the free equations which are

concentrated as t -» — oo in some neighborhood of the direction n are con-

centrated as f -* + oo in a neighborhood o f the direction — n These geomet­ric ideas are useful for understanding the choice o f 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 o f an interacting system, namely those States th at appear to be “ asym ptotically free” in the distant past and/or the distant future T o be explidt, suppose th at 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 C auchy d a ta for som e partial differential equation (acoustics, optics) O ne is interested in pairs < p _ , p> e E so that

lim ( 7 ; p - T J°V _) = 0

t — - 00

for some appropríate sense of limit, and similarly for pairs th at approach

each other as t -* + oo O ne requirem ent th at 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 th at som e 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,” th at 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 term s of free States, one m ust know th at each free State is associated

with a unique interacting State; th at is, given p there is at m ost one p such

th at T¡0)p - - T ,p -> 0 as í-> - o o N otice th at this is distinct from the re- quirem ent on the limit above th at there should be at m ost one p for each p.

(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 som e state p_ O n e hopes th at for largeposi-

tive times, the interacting state will again look like a free state in the sense

th at 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 th at the two subsets of E

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 Sim- ilarly, Í2~ is defined by lim,_ + a0 (T,(Q~p) — T \O)p) — 0 Cl* (respectively,

Q " ) is a bijection from E on to E in (respectively, E^,,) W eak asym ptotic

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 th at 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 asym ptotics of

interacting histories The reader should be w am ed th at the m aps

S' = £2+(í2- )~ ‘ : Ein-^Eom and also the m aps ( í l +) -1 íl~ and í l ” ( í l +)_1

occasionally appear in the literatura W hen weak asym ptotic completeness

holds, S' = £1" S(Q ~)" *, so S and S' are “ similar.” F o r this reason, the choice between S and S ’ is to some extent a m atter of personal preference W e 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 transfor-

mation and is called the S-operator or occasionally the S-matrix

(5) Reduction o f S due to symm etñes In m any problem s there is an underlying symmetry of both the free and interacting dynamics This allows one to conclude a priori, w ithout 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 o r schematically, S describes the response

R of a system to some input / in the following form:

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 argum ent that is intuitively in the back of physicists’ minds when discussing analytic proper- ties Unfortunately, the proofs of these properties d o not go along such simple lines We shall restrict our detailed discussion of analyticity to the two-body quantum-m echanical case (Section 7) and to the Lax-Phillips theory (Section 11)

(7) Asymptotic completeness C onsider 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 m any 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

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 asym ptotic com ­pleteness implies weak asym ptotic completeness We remark that implicit in

- ao

( 0

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

We emphasize that the above description is schematic In each physical theory there are complications, and various modifícations m ust be made Among these are: (i) In classical mechanics Z comes equipped with sets of measure zero and the natural interpretation of statem ents 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 -m echanical systems one can define an S-operator even w ithout weak asym ptotic completeness (see Section 4) Weak asym ptotic completeness then becomes equivalent to the unitarity of S (iv) In certain very special cases the free dynam ics 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

“ outg o in g ” subspaces

Usually, the«iinteracting dynamics is obtained initially by perturbing a simple dynam ics 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 asym ptotic behavior of the complete interacting system in terms of the interactions of these simple Solutions M agnon scattering (Sec­tion 14) and the H aag-R uelle theory (Section 16) are examples of such systems, as is the scattering theory for the Korteweg-deVries equation, which we do not treat

XI.2 Classical p artid a scattering

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 m otion of

r i2 = 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 dynam ical transform ation is given by T¡0)<r, v ) = ( r + vi, v) Thus the

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

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 P roblem 1) T he only place where the conditions (3) enter in the theory th at 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 th at:

| F (r) | < ,C r~ a for all r and som e a > 2 (4a)

| F(x) - F (y )| < D r~f |x — y | for all x, y with

x, y ^ r and some 0 > 2 (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 condi­tions were also required there for uniqueness This is no t surprising since according to our intuitive picture in Section 1, scattering Solutions can be

viewed as Solutions obeying “ initial conditions a t t ■» — oo.”

The conditions (4) do no t include the im portant case of C oulom b scatter­ing where the theory m ust be modified We discuss this case in Section 9

Henceforth we shall d ro p the boldface notation for vectors except in the statem ents of theorem s and in situations where confusión might arise be­tween 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 R 3 to R3 obeying (3) and (4) Let( r R6 be given with v.*, Then there exists a unique solu-tion of (2a) obeying

Proof Since we 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

H ere C, a, D, fi are the constants in condition (4) Now suppose th at 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 th at |r(r)| ^ ¿ | t | | By (4a),the integral J * , ^ ( r * , + x + m(t))| dx ds converges absolutely.

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

on (~oo, T') By the uniqueness of Solutions with initial conditions at

- T — 1, M| = u2 on ( — oo, T) |

We now define two im portant m aps:

D e fin itio n Let Z = R6 and let be the solution of (2a)

asymp-totic to a + bt at - o o Set Z 0 = ^>1^ = 0}- Then the wave operator

í l +: I 0 Z is defined by

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 d ata for a

solution of the interacting equations of m otion which is asym ptotic at

t = - oo to the solution of the free equations of m otion with d ata at t = 0

equal to w

The wave operators have several im portant properties:

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 € Z 0 ,

n * w - lim 71, T \0)w

t— * ao

where the limits are uniform on com pact subsets of £ 0 •

(b) fi* T[0) = T, Q* on Z 0 for all s.

(c) (isometry of Q * ) If F is conservative, that is, if F * — VK for some

function V, then Q* are measure-preserving transform ations.

(d) 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.

(e) If F is C°° and

3,a|F(r) ^ n r -w - 2 -«

dr\' - d i ? for all r, a and some e > 0, then Q* are C°° maps.

P roof (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¡)- ‘ = 7 1 ,, (a) is intuitively ex-

pected We shall prove the formula for Ó + ; the proof is essentially identical

for Q “ F o r fixed I e R , define J t T as before F o r (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 sam e form with t = - oo O ne now proves the

follow-ing three facts (Problems 5, 6):

(i) F o r any com pact 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 « - r» IL ^ V¡« - »II® 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, Fm q) <> 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 the fixed point of t ■ We conclude that

lim , * uí,',V r = N o w , using the fact that T_r+1 is continuous from

E to I , we conclude the proof o f (a):

(b) This is a general consequence of (a) since

0 * 7 1 ° ^ = lim 7Lf 7<0¿ w « lim 71t+ a7<0)w - T.f}**

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 E 0 x * ^ r into J í T

(Problem 7) By a general theorem on sm oothness of fixed points o f contrac- tions (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 m easure 0 In general, the range of £1* will not be all of £ o r even E m inus 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 i n i t i o n Let £¡„ = Ran Q +, = Ran Cl , and lct L bMjnd be the set

of (r, v ) so th at the solution r(t) of (2) satisfies

sup | r(f) | + sup | r(f) | < oo

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 th at Z¡„ = Z ou, = £ \ £ bound Since we have already

throw n 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 m easure 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

s u p ,|r(f)| < o o

Theorem XI.3 (asymptotic completeness; two-body classical particle scat­tering) 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 m easure 0

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

► ± 00

We first want to show th at N + and JV_ agree up to sets of m easure 0, that is,

/i(jV+yV_) + p(N _\JV + ) = 0 w herep is Lebesgue measure The m easurabil-

ity of sets like N + , N - , £(*,„„<! is left to Problem 10 Let {K„} be com pact 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 7j<q, v ) lies in a com 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 W e first note

th at p|¿°-1 Tk <= N*"* and that N l? => T, => T2 N (? =>••• Thus

p(jv?\A tf?) < 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 th at 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 ew ton’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

where r ( í) = |r( t) | and r(í) = dr/dt (which is not equal to | dr/dt | in

general) Also,

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 th at r(t) > R 0 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 th at r(í) > R 0 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 th at F(r(t)) dt

exists, so we can define

Thus, if E > 0 and l i m , ^ | r(t) | = oo, then r(í) is a scattering solution, that

is, <r(0), r(0 )) is in

Now, let L' be L with two sets of m easure zero removed: namely,

N * A N ~, which has m easure 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 m easure 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 =

F i g u r e 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),

E 0(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

Conditions (a) and (b) allow us to reduce S to a vector-valued function of only two variables F o r the family of sets {R7'J0)w |f 6 R, R e S0(3)} foliates

X into a two-param eter family of four-dimensional m anifolds (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 two-

dimensional 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 m ay suppose that r • v =» 0 o r 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 m om entum , r' and v'

lie in the y, z plañe and the com ponent of r' perpendicular to v' is

determined There are thus tw o functions that describe S: the scattering

angle 6 = arccos(¿ • z) and the time d e b y T — r' • e/p These are written as functions of the m om entum p and impact param eter b, o r 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 o r the reference in the N otes):

F i g u r e X I.2 Central scattering.

where r0(t, E) = sup{r| F (r) + f 2¡2r2 > £} and R0 is any num ber larger than ( ¡ y f í É and r0

Notice th at if V — r ~ 1 is substituted in (7a) and (7b), the integral for T

diverges but the integral for 0 converges This remark will play an im portant role when we discuss C oulom b scattering in Section 9

Finally, to m ake contact with physical experiments, we m ust define the cross section and its relation to the scattering angle 0 Let us retum to the S-transform ation in the general situation and consider a slightly different reduction from the one we discussed above W rite S (r, v) = (f(r, v), g(r, v)) We shall consider only g(r, dz) We thus “ throw aw ay ” the inform ation in / which, in term s of ou r above analysis, is equivalent to

ignoring the time delay Suppose v # 0 The relation ST¡0) = T \0)S implies

that g(r, v i) = g(r + a i, vi) for any a e IR; thus we consider only g(r, vi)

when r • z = 0 By conservation of energy | g | = v, so ¿ = g/t> We have singled out the function g(r, vi) Fix v g is then a m ap from IR2, the plañe orthogonal to i^to the unit sphere S 2 Lebesgue m easure on IR2 then induces

a measure a on S 2 by

where p is Lebesgue m easure on IR2 and £ is a Borel subset of S 2 a is called

the cross-section measure on S 2 In m ost cases, a is absolutely continuous with respect to the usual measure Q on S 2 when the forward direction 0 = 0

is removed Thus

do

do — — dQ

d íl for a function do/dQ on S 2 called the dífferential cross section.

Physical scattering experiments are well described by the following model: A beam of constant energy is sent tow ard the target The beam has a

wide spread and an approxim ately uniform density p of particles per unit

area of the plañe IR2 orthogonal to the beam A detector sits at som e scatter­ing angle <0, <p> far from the target and collects (and counts) all particles that leave the target within some angular región of size AQ about (0 , <p) The measured quantity is

num ber of particles hitting detector

(Á ñjpThe reader should convince herself, th at if AQ is very small, and the detector and source of particles are very far from the target, this quantity is very

cióse to do/dCl W e also note that there is a formula for (do/dQ)(80 , <p0) in the case where F = — V F with V(r) a function of | r | alone, in term s of the

scattering angle 9 as a function of E and b, the im pact parameter Explicitly

(Problem 12),

in the case where the sum is finite

XI.3 The basic principies of scattering

in Hilbert space

Quantum dynam ics is described by a unitary group on a Hilbert space Also, as we have seen in Section X.13, the dynamics of classical wave equa- tions can be naturally reformulated in terms of unitary groups F or this reason, the set of basic problem s and principies th at we present in this section are central to the variety of difierent scattering theories which we discuss in the rem ainder of the chapter We begin with the definition of the generalized wave operators and describe the elementary “ kinematics ” asso- ciated to that notion The existence of the wave operators is proven in most cases by a general technique known as C ook’s m ethod, which we present next Under suitable conditions that are usually m ore stringent, one can prove

existence and completeness by a complex of ideas associated with T

Kato and M S Birman C ook’s m ethod and the K ato-B irm an theory are the two pillars upon which the abstract time-dependent theory rests In concrete cases one needs technical tools for showing that the hypotheses of these methods hold—some of these tools are discussed in Appendices 1 and

2 to this section We end the section with a brief description of some of the ideas in the two Hilbert space theory and the corresponding K ato-B irm an- type theorem

Consider two unitary groups e ~ iAt and e~ iB\ which we think of as an

interacting dynamics and a com parison “ free” dynamics W hat does it mean

for e~iAt(p to look “ asym ptotically free” as t -► — oo? Clearly, it means that

there is a vector </>+ such that

lim ||e ~ iBt<p+ - e~ iAt(p\\ = O (9)

»-* - 00Notice that (9) is equivalent to

lim || eiA,e~ iB,<p+ — <p|| = 0

existence of strong limits In most applications B has purely absolutely

continuous spectrum ; but in cases where it does not, we need to choose q> +

in the absolutely continuous subspace for B F or example, if q>+ were an eigenvector for B, then the strong limit above would exist only if q>+ is also

an eigenvector of A with the same eigenvalue (Problem 15) We therefore

define the wave operators by first projecting onto the absolutely continuous

subspace of B When we discuss completeness, it will be clear that this is a

very ele ver choice!

D e fin itio n Let A and B be self-adjoint operators on a Hilbert space J f

and let P ac(P) be the projection onto the absolutely continuous subspace of

B We say that the generalized wave operators Q ±(/4, B) exist if the strong limits

Q ±(A, B) = s-lim eiAte - iBtP>c(B) (10)

l -* T 00exist When Q*(y4, B) exist, we define

J f ín = Ran Q + and J f oui = Ran Q “

F o r notational convenience, we sometimes use for and for

^ou*

The strong limit in (10) turns out to be the right one to take In case

P ac(B) = 1, the norm limit exists in (10) only if A = B (Problem 15) O n the other hand, as we shall see, if A has purely discrete spectrum, the weak limit

in (10) exists (it is 0) even though A and B are very dissimilar.

The funny convention that t -* + oo corresponds to O* is taken from the

physics literature and is connected with the relation to the “ time- independent th e o ry ” : As we shall see in Section 6, Q + is related to

limcio (x + is — ¿l)-1 and ^ to lindel o (x ” — ^ ) ~

l-The following proposition makes it clear that irrespective of its physical

im portance, scattering theory is a useful tool in spectral analysis—for this reason parts of this chapter and C hapter XIII are intimately related

P ro p o sitio n 1 Suppose that Q ±(/l, B) exist Then:

(a) O 1 are partial isometries with initial subspace P ac(B )j^ and final sub-

spaces J f ±

(b) J tf± are invariant subspaces for A and

Q ±[D(B)] c D (A \ A & i A , B ) = C1± (A, B)B (11)(c) c Ran PñC(A).

Procf (a) If m 6 [ P ^ B p f ] 1, then clearly Q ±m = 0 I f u e P c(B )jf, then

l ¿ A' e - ‘*Ptt( B M - H | for all t, so |f t* ( /l, B)u|| = ||«||.

(11) follows from Stone’s theorem and (12) F rom (12) it is clear that are

invariant subspaces for e~ iA’.

(c) By (a) and (b), A f J f ± is unitarily equivalent to B f Ptc(B )jP where the unitary equivalence is given by Q* : Pae(B )A f -* J f ± Thus A \ is purely absolutely continuous |

In quantum theory, where A and B are energy operators, (12) has an

interpretaron as energy conservation; see Section 4

The following is often useful:

Proposition 2 (the chain rule) If Í2*(i4, B) and Q±(B, C) exist, then

£1* (A, C) exist and

Q ± (/4, C) - B p ±(B, C) Procf By Proposition le, R an £i* (B, C) c R an Ptc( B \ so

üm ||(1 - Pte( B ) ¥ t* e -‘,cP K{C)(p\\ - 0 f-* T 00

for any q> Thus,

é ,Ae - i,cP"{C)q> = f ,Ae - " BP K{B )é'Be - ,,cPK{C)Q

+ e“Ae ~ i,B(l - P c(B))e,,flí’- " cP c( C >

converges to £l*(/4, B) £1* (B, C)q> as t -» + o o since a product of strongly

convergent families o f uniformly bounded operators is strongly convergent |

As discussed in Section 1, weak asym ptotic completeness says th at

J f i„ = , while asym ptotic cofnpleteness says JP¡„ = ==

[Ppp( A ) j f y where P pp is the projection onto ^ pp, the span of the eigenvec- tors of A F o r the abstract theory, a notion intermedíate between these tw o is

are com plete and ati„t (Á) = 0 Since the latter statem ent is purely spectral,

it is m ost naturally studied in a context partially disjoint from scattering theory W e discuss it in C hapter XIII

The following rem arkable fact reduces completeness to an existence question:

Proposition 3 Suppose that Q±(/l, B) exist Then they are com plete if and only if Í2± (B, A ) exist.

«

Proof Suppose th at both Q 1 (A, B) and f t * (B, A ) exist Then, by the chain rule, P c(/1) = Q ± (A, A ) = n * ( /t , B)Q± (B, A), so

P ^ A ) ^ c Ran £2* (A, B) Since we already know that Ran £2* (A, B) <= P ac(/4 )J f, completeness holds.

Conversely, suppose that £2±(/l, B) exist and are complete Let

<p e Ptc( A ) j f Then there is a ^ with <p = Q ±(>4, B)(¡/ By ou r discussion at

the beginning of the section, this implies that

||e~ ‘Á,<p - e ~ tmPtc(Byj/\\ -» 0

as t -♦ — oo Since e~ iBl is unitary,

limr-* - oo

exists and equals P ac(B)t¡> |

At fírst sight Proposition 3 seems to say that completeness is no harder than existence In fact, usually completeness is much harder The reason is that in applications B, which is the com parison free dynamics, is “ simple,” typically a constant coefñcient partial differential operator (or pseudo-

differential operator) W ith the resulting explicit formulas for e~ m one easily

shows th at £2±(/4, B) exist by C o o k ’s method Since one does not have

explicit formulas for e ~ iA\ it is not easy to s h o w f í ±(B, /í) exist Proposition

3 does suggest that one seek some condition on A and B which implies that

Q ± (A , B) exist and which is symmetric in A and B for then this condition will imply that both Q ±(A, B) and Q ±(B, A) exist, so Q ±(/t, B) will exist and

be complete This is the mechanism by which one obtains completeness in the Kato-Birm an theory

Cook’s method is based on the observation that i f / i s a C 1 function on

with/ ' € Ü(U), then lim ^ ^ f ( t ) exists since

(a) For | í | > T0 , e~ iB,<p e D(A);

0>) J?, [||(B - A ) e - i8tcp\\ + ||(B - A y iBtcp\\] dt < oo (13)

Proof Let <p e and let r¡(t) = eiA'e~iB,(p Since e~ UB<p e D{A) n D(B) for

t > T0 , r¡(t) is strongly differentiable on (T0 , oo) and

rj'(t) = - A ) e - iB,<p Thus for t > s > T0 ,

M ') - »/(s)II ^ í lk'(«)ll du < í II(B - A )e~ tB“<p\\ du

goes to zero as s -» oo by (13) Thus rj(t) is C auchy as t - »oo, so

lim^a, eiA'e~iB,Pic(Byi/ exists for all <¡/ e 3> The limit also exists trivially for all \¡/ e [ P ^ B 'p f ] 1 and, so, by hypothesis for <J/ lying in a dense set Since the family eiÁ,e~ imPtc{B) is a family of uniformly bounded operators, the exist­ ence of the limit for a dense set of ¡p implies the existence of the limit for all \¡/

by an e/3 argument This proves that íí~ exists The proof for £2+ isidentical |

In applications, one needs to control ||(B — A )e~iB,(p\\ When B is a con-

stant coefficient differential operator, this can often be done by the m ethod

of stationary phase (see Appendix 1)

In some cases one wants a variety of extensions of this theorem The

following is useful when B — A has various “ local singularices see Section 4:

Theorem X I.5 (K upsch-Sandhas theorem ) Let A and B be self-adjoint

operators and suppose that there is a bounded operator and a subspace

3> c D(B) n P ^ B ) ^ dense in P ac(B )J f, so that for any <p e there is a T0

satisfying:

C = A (l - x) ~ (1 - x)B Suppose, m oreover, that for some n, x(B + /)“ " is com pact and th at

2 c D(BT) Theji Q ±(A, B) exist.

This result follows by a simple m odifícation of the proof of C ook’s m ethod together with a general result which appears as Lemma 2 below The reader

is asked to provide a proof in Problem 19

O ne problem with C ook’s m ethod is that it requires B — A to be given to

us as an o perato r rather than a quadratic form The following result handles the form case:

Theorem XI.6 Let B be a positive self-adjoint operator and let C 0 , , C„, D0 £>„ be closed operators obeying:

(i) D(C¡) n D(D¡) => Q(B) for i = 1 , , n, and

||C¡<p||2 ^ a¡(q>, B<p) + P¡ ||<p||2, \\D¡<p\\2 < a¡{<p, B<p) + ||<p||2for all <p e Q(B).

(ii) C 0 = 1, Q(D0) o e(B ), and

for all <p e Q(B).

(iii) The quadratic form ]¡T?=0 CfD, defíned on Q(B) is symmetric and

(iv) There is a set 2 contained in Ran P ,C(B) n D(B) which is dense in Ptc(B),Wy so th at for <p e 2 ,

(a) for | r | > T0 , (1 - x)e ,BV e D{A);

Then the form sum A = B + Y j - o C fD , is a self-adjoint operator and n*(A , B) exist.

Proof By (i), (ii), and (iii), CfD¡ is a relatively form bounded pertur- bation of B with relative bound a = j j - o < 1 It follows th at A is self- adjoint and that Q(A) — Q(B) In particular, the norm s

on Q(B) are equivalent norm s, th at is,

c j | | 9 | | , á M U ^ M U

Here £ is some fixed num ber so th at A + E ^ 1 e ~ im is clearly an isometry

in || • ||B Since e~u ' is an isometry in || • ||M, by the above equivalence, we

We shall prove that as t, s -* 00, each of these term s goes to zero, so that, as

in Cook’s theorem, fl* (A , B) exist W e consider the first term ; the second is

similar We first claim that

m t) c p , (WK(f) - W(s))<p) = <{' ¿ ( C je - ‘A*W(t)<p, D je~ iBu(p) du (15)

is a rank one operator, th at is, (B — A)tp — (i¡t, <py¡/ If we tried to use C o o k ’s

m ethod to show that Q ±(/4, B) exist, we would seek <p with (^, e~ itB<p) 6 I?(R) Since <p e P iC(B )jf, we know th at the spectral m easure d(<p, E x q>) equals |/( A ) |2 dX for some f We shall see below that it follows th at d(^, E x <p) = g(A )|/(A )|2 dX for some g in L2( R , / 2 dX), and thus

(*, e - “B<p) = ¡ e - ^ g ( X ) l / W ¡ 2 dX Therefore, (ift, e~ i,B<p) is the F ourier transform of (2n)1,2g | / 12 In general, it is not easy to see when a F ourier transform is in I } but to get it to be in I? is easy W e therefore begin by finding a set of <p with (\¡/, e~ l,B<p) e L2(R).

Definition Let B be a self-adjoint o p erator and {Ea} its spectral family

J f( B ) will denote the set of all (p e such th at d(<p, E x<p) = |/ ( A ) |2 dX

w here/ e L°°(R) We let |||<p||| be the L°°-norm o ff

It is not hard^P roblem 17) to prove th at ||| • ||| is a norm and th at J ( ( B ) is

dense (in the ^ f-n o rm ) in Ran P ,C(B)

Lemma 1 F o r any <pe J t( B ) and any i¡te Jf,

f | W , e - » V ) l 1 * £ 2 » | | * | ! |W ||> (16)

Proof Let Q be the projection on to the cyclic subspace generated by B and

<p Let d(<p, E x q>) = | / (A) |2 dX By general spectral theory (see C h apter VII and Section VIII.3) Q j f is unitarily equivalent to L2(R, | / (A)|2 dX) in such a way th at <p corresponds to the vector <p(X) s 1 and e~',B is m ultiplication by e~ia Let r¡(Á) correspond to the vector Qip Then

(*, e - " V ) = (G^, * " 'V ) = { f/(A )|/(A )|2e~ ',A dX (17)

so, by the Plancherel theorem,

f dt = 2 n ¡ M A ) | 2 | / ( A ) r d A

^ 2 n | | / | | 2 { \ n(X)\2 \f( X ) \ 2 dX

By definition 1/11* = |||<¡p||| and

We shall need another simple consequence of thinking of the unitary group in terms of the F ourier transform :

L em m a 2 F o r any <p e P ,e( B \ e~',B<p-*0 weakly as t -* ±00 If C is compact, then ||O e- *r®<p|| -» 0 as t -* ± 00

Proof By (17) and the fact th a t/ and rjf are in I?, we have th at (\¡/, e~ i,Bq>) is

the Fourier transform of an L1 function So, by the Riem ann-Lebesgue

lemma (Theorem 1X.7), (i¡/, e ~ i,B<p) 0 Thus, ||F e _,,B</>|| -» 0 for any finite

rank operator F The result for com pact operators follows by an e/3

for all (p 6 J((B ) We shall prove this by writing the left-hand side as two

pieces, one to be controlled by Lem m a 1 and one by Lem m a 2 Let

r »

* * ( * ) - [ eiB,X e ~ i8 ,dt

a

for a bounded operator X and a < b We first claim that

lV(r)*VF(s) - eiaBW (t)*lF (s)e-i,ir = F0.(Y (t, s)) (19)

Y(t, s) = - i [ e i,BJ* e ~ iu- s)AC e - lsB - ei,BC*e~M- s)AJ e ~ i,B]

We shall prove (19) w ithout worrying about dom ain questions, leaving the reader to take m atrix elements and fill in these dom ain details T he idea will

be to write the diflerence on the left as the integral of its derivative Let

Q(b) » ¿ bBW {t)*W (s)e-ibB

Then

_ ie'hB[Be‘,BJ+e ~¡(t-s)AJg-lsB _ e¡tBJ0e ~Ht-s)Aje ~tsBgje -¡bB

= i¿ bB[ei,BJ* e~ i{,~*)AC e~i,B — ei,BC*e~l*'~**AJe~'*B]e~ibB

is com pact, so by Lem ma 2,

lim eiaBW (t)*(W (t) - W (s))e~iaB(p - 0

a-* oo

for (p 6 It follows by (19) th at for <p e

(q>, W(t)*(W(t) - W(s))cp) = lim (<p, F0a(Y(t, t) - Y(t, s)>p) (20)

a-* oo

Since C is trace class, it has an expansión (see (VI.6)):

c =

m = 1

where £ X„ — |j C71| t , the trace class norm of C, and with {(/>„} and (i¡/m}

orthonorm al and > 0 We claim th at for any bounded op erator X and

26 XI: SCATTERING THEORY

For, by the canonical expansión above,

and, in the second place, since £ » A, | (</>* e~ ixB<p)\2 is in L‘, (18) follows |

As a corollary of the theorem and (23), we have:

C o ro lla ry U nder the hypotheses of Theorem XI.7,

Proof In (23) take s = 0 and let t -* ± oo |

If A J - JB is trace class, then so is BJ* — J*A , so both s-lim ¿ Á,Je~ imPK(B) and s-lim e‘mJ* e~ iA,Pme(A) exist F o r general J , this

does not imply completeness of either strong limit (for example, consider

J = 0); but if J = 1, Proposition 3 is applicable, so we immediately have the

In this theorem A and B may be unbounded A — B trace class is intended

in the sense of Theorem XI.7, th at is, (A<p, i//) = (</>, B>p) + (<p, C\j/) for som e

C e J x and <p e D(A), \¡/ e D(B) It then follows that D{A) = D(B) and A<p = B<p + C(p for <p e D(/4).

Corollary Let {/!,,}" lt A, B be self-adjoint operators Suppose that Íl*(y4, B) exist and that each A„ - A is trace class with ||/t, — >4||, -+ 0 as

n -* oo Then, for each n, Q* (Am, B) exist and

n * (i4 , B) = s-lim n * ( y í.,B )

as n -» oo If ÍI*(B , A ) exist, then for each n, Ci±(B, A„) exist and

Q*(B, A)q> xs lim Q*(B, A„)q>

«-♦00

for all <p 6 Ran Ptc(A).

Proof By the chain rule, it suffices to prove that

«“♦00and

«“♦00

From the corollary to Theorem XI.7 we immediately conclude that (25)

holds Let <p be in Ran P c(/4)andlet</>H = íl* ( A H,A)(p By (25), ||</>„ - <k>|| -» 0

as n -+ oo Thus,

||í l +( / l , / l B)(<pB- « p ) H 0

But, by the completeness of Q +(/ln, A), we have that 0 +(/4, A m)(p„ = q>,

so the last limit result says th at (26) holds |

It can happen th at for <p e [Ran P ,C(A )]x, Q* (B, A„)q> does not go to zero

as n -> oo (Problem 22).

O ne cannot replace the trace class condition in Theorem XI.8 by a condi­

tion th at A — B be H ilbert-Schm idt o r that A — B be any J p with p > 1; see

the discussion in the Notes O ne problem with Theorem XI.8 is th at in

quantum mechanics B - A is not even bounded.

Theorem XI.9 (K uroda-B irm an theorem ) Let A and B be self-adjoint operators so th at (A + i) ' 1 - (B + i)" * € / , Then f l ±(/l, B) exist and are

complete

Proof Let J = (A + i) l (B + i) Then, in the sense of expectation valúes,

A J — J B = (B + i)-1 — (A + i)-1

is trace class, so by Pearson’s theorem

s-lim eiA'(A + i) ~ 1{B + i )~l e ~ iB'Pac(B)

exist Applying this to a vector of the form (B 4- /)</>, we conclude that

Ü ± (/l, B) exist It follows by symmetry that Í2± (B, A) exist and thus com­

pleteness holds |

To state the next result, we need a technical definition:

D e f in itio n Let A and B be self-adjoint operators We say that A is subordínate to B if there are continuous functions/ and g on IR w ith/ (x) > 1,

0( x ) > l , and lim|xH oc/ ( x ) = oo such that D(g(B)) c D( f ( A ) ) and

f ( A ) g ( B )~ 1 is bounded If A is subordinate to B and B is subordinate to A,

we say they are mutually subordinate

This condition is very weak F o r example, by the closed graph theorem , if

D(A) = D(B) or if A and B are semibounded and Q(A) = Q(B), they are

mutually subordinate

T h e o r e m XI.1 0 (Birman’s theorem) Suppose that A and B are self- adjoint operators with spectral projections Ea (A), £ n (B), respectively

Assume that:

(a) Ej(A)(A - B) E¡(B) e J x for every bounded interval /.

(b) A and B are mutually subordinate.

Then Ü ± (/I, J3) exist and are complete

Proof By symmetry and Proposition 3, it suffices to show that íí* (A, B) exist Let Ea(C) = £ (_a a)(C) and E'a(C) = £ , - » , - al u[a (C ) where C is A

or B I Í J = Ea(A)Ea(B), then A J - J B e by hypothesis (a), so

s-lim eiA'E„(A)Ea( B) e- iB‘

exist by Pearson’s theorem Let cp e Ran E ao(B) for some a0 Then for

a > a0 we have that

lim eiÁtEa(A)e~iBtq>

X-* ± 00

exist, so to conclude that B)<p exist, it suffices to show that

Now, let/ and g be the functions given by the condition that A is subordínate

to B Let F(a) = infWí>fl/(x ) Then F(a) -► oo as a -► oo since/-► oo Thus:

||£ ;(/l)e - iB'<p|| < F ( a ) - l \\f(A)E'a( A ) e - iB'<p\\

There are a large num ber of conditions that arise in applications but which are not covered by the above considerations F o r example, suppose

that > 4 ^ 0 , B > 0 and A 2 — B 2 e J , D o B) exist? O r consider

A = — A + K; B = — A on IR" F o r n ^ 4, (A + i)-1 — (B + i)-1 is not trace class for any nontrivial V\ but, as we shall see, (A + E)~k — (B + E)~k is trace class so long as k is large enough Does this imply that Q * (A, B) exist?

The answer to both questions is yes because of the general principie which

we are about to describe

Definition A function <p on T, an open subset of R, is called admissible

if T = ( J í /„ where /„ = (a„, /}„) are disjoint, N is finite or infinite, and: (a) The distributional derivative q>" is L1 on each compact subinterval of T ; (b) on each interval (a„, fi„), <p is either strictly positive or strictly negative.