Potential scattering with confined channels

ANNALS

OF PHYSICS

102, 1-70 (1976)

Potential

Scattering

with

Confined

Channels

R. F. DASHEN* Institute for Advanced Study, Princeton, New Jersey 08540 J. B. HEALY+ Physics Department, Yale University, New Haven, Connecticut 06520 AND I. J. MUZINICH* Brookhaven National Laboratory,

Upton, New York 11973

Received June 1, 1976

We introduce a class of nonrelativistic multichannel potential scattering models which are characterized by the simultaneous presence of and communication between two distinct types of channels: First, ordinary two-particle scattering channels, the Hamiltonian for which has an absolutely continuous spectrum on the positive real energy axis and perhaps a &rite number of negative energy bound states; second, permanently confined channels (like those of the quark model), the Hamiltonian for which has only a point spectrum with an accumulation point at E = + m. These two types of channels are connected in the full Hamiltonian .%’ for the multichannel system by off-diagonal local potentials which satisfy suitable smoothness and integrability conditions. The scattering theory of such systems is developed and, under certain general restrictions on the potentials, the following properties are rigorously established: (1) Asymptotic completeness; i.e., the generalized wave operators exist and are complete, and the S-matrix is a unitary operator in the scattering channels. The S-matrix has nonzero matrix elements only in the scattering channels. (2) The spectrum of the Hamiltonian S consists of 3 parts, namely, a finite number of negative energy eigenvalues, a discrete set of positive energy eigenvalues with only possible accumulation point at E = + co, and an absolutely continuous spectrum on the remainder of the positive real energy axis. To each eigenvalue, positive or negative, corresponds a finite number of orthonormal eigenvectors, the bound states of S; and to each positive energy there * Research sponsored, in part, by ERDA Grant No. E(ll-1)2220. + Research (Yale Report No. COO-3075143) sponsored in part by ERDA Grant E(ll-1)3075. *Work supported in part by the Energy Research and Development Administration. Copyright Q All rights

1976 by Academic Press, Inc. of reproduction in any form reserved.

1

No.

DASHEN,

HEALY

AND

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corresponds a unique bounded solution, the distorted plane wave, to the time-independent SchrBdinger equation. (3) There is an eigenfunction expansion associated with 2 in which enter only the bound-state eigenvectors and the distorted plane wave eigenfunctions. (4) The subspaces of discontinuity and absolute continuity of 8 are orthogonal complements to each other. In addition the scattering amplitude for these models is constructed and shown to be related to the S-matrix in the usual way. Both the S-matrix and the scattering amplitude are well defined and continuous at the positive boundstate energies.

1. INTRODUCTION

The hypothesis that the interactions among “elementary” particles should be governed by a local quantum field theory with cotied “quarks” has achieved wide acceptance among high-energy physicists. The idea is that the quarks (which may or may not have a fractional electric charge) are the fundamental building blocks of the observed hadrons [l]. However, the interactions between them are so strong that they cannot appear in asymptotic scattering states, except in the bound configurations which make up the observable hadrons; that is, one can never observe a free quark [2]. While certain field theories in two space-time dimensions are known to possess this property [3], and some field theories in four space-time dimensions are hoped to have it, it is a fundamental problem to demonstrate that it is in fact realized in any realistic local quantum field theory. Unfortunately we have nothing to say here about the existence (or nonexistence) of such theories. Rather, we assume that confined (i.e., permanently bound) quarks exist in nature and ask what can be learned about the scattering theory of observable particles in such a situation. Since we do not have a satisfactory way to answer this in field theory, we will resort to the time honored method [4, 5] of using nonrelativistic local potential scattering as a guide. Within that context we will rigorously prove certain general properties of scattering systems coupled to confined channels, with the hope that at least some of these properties carry over to the relativistic world of high-energy physics. In the usual nonrelativistic quark model [l, 21 the quarks have spin &, the mesons are quark-antiquark bound states, and the baryons are bound states of three quarks. For simplicity we will ignore the spin degrees of freedom of the quarks and consider only mesons. We will not distinguish quarks from antiquarks, but to avoid confusion we retain the terminology. In the absence of interactions between hadrons, the allowed states of each free hadron are the eigenstates of a Hamiltonian operator H, , which in coordinate space takes the form Hdr) = -V,Y2

+ U,(r)

in the hadron’s center of mass frame. Throughout

U-1) this paper we set ?& and the

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3

quark mass and hadronic ground-state mass equal to unity; nothing new is added by allowing the quarks or hadron ground states to have arbitrary nonzero masses. We require the potential U,(r) to be such that H, is a self-adjoint operator having a pure point spectrum {CO,}, bounded below and with only accumulation point of eigenvalues at E = + w; and the eigenvectors en(r) of H, form a complete orthonormal set in L,(E), where E is the three-dimensional Euclidean space of position vectors. The new feature (other than the composite nature of the hadron) which we wish to introduce into potential scattering will be the coupling between two-hadron scattering states and confined (quark-antiquark) states, i.e., single hadrons. As in ordinary two particle potential scattering we will ignore the coupling of twohadron states to states containing more than two hadrons. Temporarily we assume that the only interactions between hadrons are those generated by the annihilation and creation of quark-antiquark pairs. With these simplifications hadron-hadron scattering can be described, in the hadronic center of mass system, by the coupled SchrSdinger equations

l?UrJ + HG(rO) + H,(R)1y,(r, , rD, R, t) +

s

dr V(r, , r, , R;

r)

Yl,(r, t) = i(a/at) YJr, , r, , R, t),

(1.2a)

where H,(R) = -(VR2/2) is the kinetic energy operator for relative motion between the two hadrons; and V(r,, rO, R: r) = (r, , r0 , R I VI r>

(1.3)

is the integral operator which describes the transition between the two hadron and one hadron states via creation and annihilation of quark-antiquark pairs. We assume that V(r, , rs , R; r) is real, but this is not essential. If I/ is complex, then in Eq. (1.2b) V should be replaced by its complex conjugate. Physically the natural way to proceed is to expand Ys(r, , rs , r, t) and V(r, , r, , R; r) in the eigenfunctions 5, of H, , which have been assumed to be complete in L,(E):

(1.4)

4

DASHEN,

Then using the orthonormality of coupled equations: W,(R) + w, + s,l ‘CV,

HEALY

AND

MUZINICH

of the [, we can write Eqs. (1.2) as an infinite set

0 + 1 dr V,,(R

r) Ydr, t)) = GW)

y?‘YR, 0, (1.5a)

H,(r) Ydr, t) + s dR ( C V,dr, 12,111

RI YFYR, t)) = i(a/at> Ydr, t>.

U.5b)

Each scattering channel, labeled by the integers IZ and m, corresponds asymptotically to a scattering state containing two free hadrons, one in the 11th eigenstate of Ho with mass w, , and the other in the mth eigenstate of H, with mass w, . To make Eqs. (1.5) tractable we assume that only a finite number of potentials V,,(R, r) are nonzero, and that these are local: V,dR, r> = vdr>

&R - r>.

(1.6a)

For simplicity of notation we will carry out our analysis explicitly for the case of only one nonzero potential V&R, r) = V(r) 6(R - r).

(1.6b)

Then we can forget about all channels with n or m > 1 and study only the twochannel Schrlidinger equation 2-f-Y = (&

+ q

where the unperturbed Hamiltonian H s andHO=H,-22w e 0

Y = (i(a/at) - 2oJJ Y,

(1.7)

XU is a 2-by-2 diagonal matrix with elements

(1.8) and the perturbing potential V is, in coordinate space, the multiplication by the real symmetric off-diagonal matrix

operator

v = V(x)(; A). The wave function Y is a 2 component column vector with scattering and confined channel elements Y, and Y, , respectively, (1.10)

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We will also study the time-independent iPP(x)

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5

SchrGdinger equation = EY(x)

(1.11)

obtained from Eq. (1.7) by taking Y(x, t) to be an energy eigenfunction Y(x,

t) = Y(x)

eci”Jt =

t

Ydx) ye(x)

e&,A >

(1.12)

with E = w -

2w,.

In the sequel we will drop the superscript on HCo, denoting it simply by H, . Its spectrum is {CL,] with pn = w, - 2~0, and p. = -w. . The Schrodinger Eqs. (1.7) and (l.lO), with only one scattering channel and one confined channel, certainly describe the simplest class of system which couple scattering and permanently confmed channels; and they are worth investigating for that reason alone. To the best of our knowledge there has been no general discussion in the literature of systems of this type. The generalization of our results to any finite number of scattering channels (i.e., any finite number of nonzero I’&, and to any finite number of confined channels as well, is trivial. We could also include diagonal terms in the interaction V; but this would make our proofs more cumbersome and add nothing new. Therefore we will confine our discussion of the possibility of a direct interaction between the hadrons to some remarks at the end of the paper. Our task then is to investigate the scattering theory of the Schrodinger operator &’ in Eq. (1.8). The restrictions on UC and V which we will use are somewhat technical, and will be stated in Section 2; here we remark that the allowed class of confining potentials includes the three-dimensional harmonic oscillator. We prove that the Hamiltonian A? is a self-adjoint operator in the same domain 9(Z) = B(JQ in which the unperturbed Hamiltonian tiU is self-adjoint; like sU, the operator &’ is lower semibounded (the only restriction on the off-diagonal potential needed for the proof of these properties is V(X) E L,(E)). We define generalized wave operators W* as the partially isometric operators W+ = S -,iirnm eitdE”e+?Y,

,

(1.13)

where 9, is the projection operator onto the scattering channel. We also define the S-matrix in terms of the wave operators by the relation s = w+*w-.

(1.14)

The only nonzero matrix elements of S are the diagonal scattering channel elements. The wave operators and the S-matrix are basic quantities in the mathematical

6

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description of the scattering process [4, 6, 71. The S-matrix is a unitary operator in the scattering channel if and only if the ranges of IV+ and W- coincide. We demonstrate that our generalized wave operators exist and are complete; and so the spectrally absolutely continuous parts of the Hamiltonian operators J? and flU are unitarily equivalent. It follows [6,7] that the S-matrix exists and is unitary (for these properties we need only V(x) E h(E) n L,(E). This is the form of asymptotic completeness which Simon [6] calls “Kate asymptotic completeness.” We also consider the eigenfunction expansions associated with the Schriidinger operator .%. For single channel potential scattering with local L2 potentials, Ikebe [S] has proven eigenfunction expansions which contain only a discrete set of negative energy bound states and a continuum of positive energy “distorted plane waves.” Other authors [6] have extended the proof to include a wider class of potentials; but as Ikebe has pointed out, there is no complete theory for eigenfunction expansions associated with partial differential operators. For our twochannel Schriidinger operators &’ we have established eigenfunction expansions in which three classes of eigenfunctions occur, namely: a discrete set of negative energy bound states, a discrete set of positive energy bound states, and a continuum of positive energy “distorted plane waves”. Let us be more precise. Let g,,(x, y, k) be the kernel of the resolvent operator 9% = (k2 - #U)-l. The homogeneous integral equation (1.15) has nontrivial bounded solutions only at a discrete set of eigenvalues k2 = En on the real energy axis. The set of eigenvalues {En} is bounded below, each eigenvalue is of finite multiplicity, and the only possible accumulation point is E = + co (in showing the discreteness of the positive energy eigenvalues we will need to restrict V to be &rite range). The corresponding solutions U,(x) are square integrable bound-state solutions to the time-independent Schriidinger equation (1.11) and will be referred to as eigenvectors; they can be chosen to be orthonormal. On the positive real energy axis the inhomogeneous integral equation

u’,(x, k) = (O1l)-Tk-),

(1.16)

k2 = E > 0,

has a unique bounded solution, except at the positive energy eigenvalues of &’ (at each eigenvalue a unique bounded solution to (1.16) is selected by the additional

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7

requirement that Y(x, k) be a continuous function of k). These eigenfunctions are the distorted plane wave solutions to the Schriidinger equation (1.11). With these eigenfunctions and eigenvectors our expansion formula for an arbitrary function t (x) E [L,(E)]2 = L,(E) 0 L,(E) is: (1.17a) where

+(k) = j-, Y+(x, 4 f(x) dx,

(1.17b)

jn = s, IY,+(x) f (4 dx.

(1.17c)

Except for the presence of the bound states in the continuum, this has the same form as the eigenfunction expansion proved by Ikebe [8]. In particular, there is no singular continuous spectrum, and Eqs. (1.17) show that the bound and scattering states are complete. Our proof of the eigenfunction expansion associated with X is patterned after Ikebe’s proof [8] for L, scattering potentials. However, Ikebe then used the eigenfunction expansion to prove asymptotic completeness. We have found it necessary to reverse the order of these steps; the assumptions on the off-diagonal potential Y needed to prove the eigenfunction expansion are much stronger than those required for asymptotic completeness; and we need the unitary equivalence of the absolutely continuous parts of the operators 2 and HU to show that in the orthogonal complement of the subspace spanned by the eigenvectors of 2 the spectral measure is absolutely continuous, even at the eigenvalues En . This in turn allows us to prove the stronger form of asymptotic completeness which Simon [6] refers to simply as “asymptotic completeness.” Finally we define the T-matrix by T(k, k’) = j-, Yf+(x, k) V-(x) Y(x, k’)

(1.18)

and show that it is related to the S-matrix in the usual way. In a separate paper [9] we investigate the partial wave Schrijdinger equations for the case of spherically symmetric potentials. We establish the analytic properties of the solutions to the partial wave equations, the relevant Fredholm determinants, and the partial wave S-matrices. The Fredholm determinants are found to have poles at the bound-state energies of the confining Hamiltonian and zeros at the bound-state energies of the full Hamiltonian; but the partial wave S-matrices have poles only at the locations of the negative energy bound states of 8. From the

8

DASHEN,

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properties of the Fredholm determinants we derive a generalization of Levinson’s theorem which is applicable to multichannel scattering systems with some confined channels: Let p, e be the energy of the nth eigenstate (counted according to its multiplicity) of H, with angular momentum 8, and assume that for large energies the level spacing B = &%+I - p,l is bounded below by a constant dmin . If S,(E) is the physical phase shift for angular momentum 4, N/(E) is the number of bound states of &’ with angular momentum / and energy less than E, and E is any real number such that ptL,in/2 > E > 0, then for large n we have the approximate equality &(O> - [Upne + 4 - rrnl - rNi’(l*.ne + c>;

(1.19a)

and the approximation becomes arbitrarily accurate as IZ + 00 with E fixed. As in ordinary potential scattering, a nonnormalizable S-wave solution to Eq. (1.15) counts as half a bound state in Eq. (1.19a). If the total number of bound states of &’ with angular momentum 4 is some finite number NBd, then our generalized Levinson’s theorem takes the simple form: S,(O) -

lii [Sef’pne + l ) - m] = TN/.

(1.19b)

We then study in detail an explicit example, namely the case of an infinite radial square well confining potential with finite square well off-diagonal potential. The partial wave Schrodinger equations can be solved exactly, and we are able to construct an explicit example of a bound-state degenerate with the continuum. The number of bound states in each partial wave is found to be finite. And in conformity with our general statements the S-matrix is not singular at the locations of the positive energy bound states. The remainder of this paper is organized as follows: Our notational conventions and restrictions on the potentials UC and V are introduced in Section 2. In Section 3 we examine the properties of the kernels SU(x, y, h) and K(h) = ??&x, y, X) Y( JJ). In Section 4 we prove “Kato asymptotic completeness” and establish some related properties of the wave operators. Section 5 takes up the question of the eigenvalue spectra of the integral operator K(h) and the Schriidinger operator X, and the equivalence of the eigenvectors of K(X) with bound-state solutions to the Schrijdinger equation (1.11). In Section 6 we investigate the properties of the resolvent W(h) = (h2 - &‘)-I as an integral operator, and in Section 7 we establish the existence of the continuum eigenfunctions and their relation to the resolvent kernel 9(x, y, X). Finally in Section 8 we prove the eigenfunction expansion as well as a stronger form of asymptotic completeness; we also introduce the scattering amplitude as a kernel in momentum space and establish its relation to the S-matrix. Section 9 contains a discussion of the possibilities of enlarging the class of potentials which we have studied and of making

POTENTIAL

9

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our models more realistic. We conclude by making some general remarks about the physics of systems in which the observable particles are bound states of permanently confined quarks. The Appendix is devoted to a detailed discussion of the properties of the Green’s function for the three-dimensional harmonic oscillator. 2. RESTRICTIONS ON THE POTENTIALS AND SOME NOTATIONS Let E be the three-dimensional Euclidean space of position vectors (x), and let M be the three-dimensional Euclidean space of momentum vectors (k). Then for any functionf(x) E L,(E) we can introduce its Plancherel transformfO(k) E L,(M); the correspondence f(x) ++ fa(W is one-to-one, linear, and isometric in the sense that

(Note: we use the notation (I I/ exclusively for L, norms.) If

dx<00, sEIf(x)!

(2. la)

then f&k) = (24-3/z j-J(x)

cik.% dx;

(2.lb)

and if sM IfoW

(2.lc)

dk < ~0,

then f(x) = (27~-~/~ JMfo(k) eik.mdk. In coordinate operator :

space the free particle Hamiltonian

H, is just the Laplacian

H, = -VZ2.

The domain D(H,) in which the scattering channel Hamiltonian operator is characterized by [7]

h,(k) E L,(M), k2fo(k) E L,(M).

(2.ld)

(2.2) H, is a self-adjoint

(2.3)

10

DASHEN,

HE&Y

AND

MUZINICH

Let us denote by R,(h) the resolvent operator of H, : &(A) se (A2 - H&l.

(2.4)

For Im X > 0 the resolvent R,(h) is well known to be a Carleman type operator which can be represented by the integral kernel

QW)f)(x) = h G&G Y, 4f(u) 4,

f E L(E),

G,(x, y, A) = -(1/27r)(e’AlZ-vl/l x - y I).

(2.5a) (2.5b)

The Green’s function G,(x, y, A) also satisfies the condition

II GA *yA)l12E sE j G&x, y, h)12dy < const

(2.6)

independent of x for each fixed h in Im h > 0. We will refer to the condition (2.6) as the strong Carleman condition. For the confined channel Hamiltonian we impose the following restrictions: H, is a self-adjoint operator in some domain D(H,) E L,(E); its eigenvalues E = ps form a discrete set in -w0 < E < co; each eigenvalue is of finite multiplicity and there are no finite accumulation points of eigenvalues. The eigenvectors {5,} of H, form a complete orthonormal set in the sense that for any f(x) E L,(E) the generalized Fourier coefficients (2.7a) exist and the norm off is given by:

Ilfll” = C 13sI2= s, I fWl” dx12

(2.7b)

In other words f(x) E L,(E) implies 3,, E t2 . For all f E D(H,) the range of H, is also contained in d2 :

IIW-II2 = 1 pn2I3nI2-=c00. n

(2.8)

Let R,(h) denote the resolvent operator of H, : &(A) = (A2 - H&-l.

(2.9)

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We will require that for X2 # CL,,the resolvent R,(X) is an integral operator of Carleman type which can be represented by an integral kernel G,(x, y, h): 0-U) f>(x) = s, Gck Y, @f(v)

f E ME);

4s

(2. IOa)

and the right-hand side of Eq. (2.10a) is a Holder continuous function of x. Using the eigenfunction expansion associated with H, , we have the alternative representation: (2.10b) (W) f)(x) = c &(-4/W - cJ)(yI, 2f>* n For fixed P # pn, G, must satisfy the strong Carleman condition (2.1 la)

II Gdx, ‘3 x>ll -=I C, the asymptotic condition

(2.11b) and the bound

I Gob,Y, &I < C/l x - Y I.

(2.1 lc)

And in the limit of large negative energy,

2; IIG,(x, *, A)ll- At-m 0.

(2.12)

The class of potentials U, for which the Hamiltonian H, satisfies our conditions is not immediately obvious. We do know that the admissible confined channel potentials include both the three-dimensional harmonic oscillator (this is proven in the Appendix) and the three-dimensional infinite square well potentials (i.e., infinite square well in the radial coordinate or infinite square well in each of a particular set of Cartesian coordinates; the proof is trivial). We define Z&Q to be the domain in (L,(E))2 E L,(E) @ L,(E) consisting of D(H,) in the scattering channel and D(H,) in the confined channel %%J

= Wfs) xa + W&)

where

xc = (33 xc = (4.

xc >

(2.13)

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DASHEN, HEALY AND MUZINICH

The unperturbed Hamiltonian and the resolvent operator

XU is a self-adjoint operator in the domain B(&);

W,(h) = (A2 - .3fp

(2.14a)

can be written as (2.14b) For the off-diagonal potential we assume throughout

that

V(x) is locally Holder continuous except possibly at a finite number of singularities; and V(x) decreases at least as fast as O(l x 1-5/2-E)for large ) x 1, i.e., there exist positive constants C, R, and E such that

\ V(x)l < c - 1x 1--6/2--E, 1x1 >R.

(2.15b)

To fully investigate the nature of the positive energy spectrum of the full Hamiltonian &’ we have found it necessary to introduce the additional assumption that V(X) is of finite range, i.e., there exists an q, > 0 such that for 0 < 01 < q, ealzlV(x) E L,(E).

Whenever we use this last assumption it will be stated explicitly.

3. THE INTEGRAL

OPERATORS ~&,y,X)

AND K(h)

In proving that the strong Carleman condition on the confined channel Green’s function G,(x, y, A) is satisfied for particular confining potentials and in using this condition it will be useful to relate it to the eigenfunction expansion associated with H, . LEMMA

3.1. For fixed X2 # pu, and$xed x, the L, norm of G,(x, y, A) is given

by lliG&

-9U”

= c (5,“Wl A2- Pn I”); n

(3.la)

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CHANNELS

and G, satisfies the strong Carleman condition, Eq. (2.1 la),. fund

only if

(3.lb)

c G!n2(x>/l A2 - t-h I”> < c,

9.

where C depends on X but not on x. Proof.

We have only to prove Eq. (3.la). By Eqs. (2.7) and (2.10)

(3.2)

where we have used the orthonormality

of the f,(x).

Q.E.D.

Next we will establish that the potential V(x) is Kato tiny (in the terminology of Simon [6]) relative to XU, and so can be regarded as a small perturbation of the unperturbed Hamiltonian. That is we will show that for YE 9(HU), II VYll

G a II KYII

+ b II W,

where b > 0 and the positive constant a can be chosen arbitrarily

(3.3) small.

LEMMA 3.2. For any YE SB(AQ the component functions Y8(x) and Y,(x) are bounded and the potential V(x) is relatively bounded with respect to SU with relative bound 0. Proof. Since V(X) E L,(E) and [V(X)]~ = V2(x) I, it suffices to show that all Y(x) E 9(tiJ are bounded, with

II Ydx>llm < a, II HsYs II + b, II Y, II, II ~c~~)llm G a, II fWc where a, and a, can be chosen arbitrarily proof is standard [7]:

II + b, IIYc II,

(3.4)

small. For the scattering channel the

II Y’, Ilm G CW-3’2 1 dk I %k)I < [s @WI k I2 + a”>“>j” (I k I2 + a2j2 I ?k,(k)l” dk]“’ = (T”/W”

IIW, + a”) Ys II < (7+“3(274-“‘“[!I

H,Y, II + o? II Ys II].

(3.5)

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For the confined channel we proceed analogously, replacing the Fourier transform with an expansion in eigenfunctions of H, :

< f(4U W’c II+ a2IIyc III,

(3.6a)

where

f”(4 = 11 c (I Lz(X)12/(a2 + PTJ2)11 m n (3.6b) and we have used Lemma 3.1 and the assumption Eq. (2.12) on G, . Thus by choosing (Yas large as we please in Eqs. (3.5) and (3.6) we can make the constant a in Eq. (3.3) arbitrarily small. Q.E.D. Now let us proceed to the analysis of the resolvent operator W,(h). That W,(A) is a Carleman operator and gU(x, y, A) satisfies the strong Carleman Condition for Im X > 0 is obvious; we have then LEMMA 3.3. Forpxed h in Im h > 0, A2 # cc,,, the resolvent W,(A) is a Carleman type operator with kernel 9JU(x,y, A):

The Green’s function ??Jx, y, A) satis$es the strong Carleman condition

II ~u(& *, A)/]< const * z,

(3.7b)

where the constant is independent of x (but not A). LEMMA

3.4.

The potential Y can be written as a product of two factors Y- = ‘IGr’T

such that both V’(XU - y)-l and Y”(& in (L2(E))2 for real y < 0, y # pLo.

- y)-l

(3.8) are Hilbert-Schmidt

operators

POTENTIAL

Proof.

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15

Choose

Both V’“’ and V’ are E(L,(E))~ (since VE L,(E)), and since 9, satisfies the strong Carleman condition for Im h > 0, X2 # p,, (in particular for X = i(j y \)l/3 the kernels V’(x) SJx, y, i(l y j112) and V”(x) SU(x, y, i(j y j)1/2) are both HilbertSchmidt. This proves the theorem, since SU(x, y, i(l y 1)1’2) is the kernel of (y - ZU)-l as an integral operator in (L2(E))2. Q.E.D. It will be useful to extend the operator R, . We know that the resolvent R, is Carleman type for Im A > 0 and has the representation Eq. (2.5) when operating on functions f(x) E L,(E). R, is not Carleman type for Im X = 0, but we can use the representation (2.5) to define an extension T,(h) of R,(h) to Im X = 0 and to functions outside of L,Q:

for Im h > 0 and all g for which the integral exists. This in turn allows us to define an extension of W, to Im A 3 0, X2 # pL,, since R,(h) is already well defined in that domain. We have defined our Hamiltonians as operators in Hilbert space. In order to apply time-independent scattering theory methods it will also be useful to introduce a space of bounded functions. Let B be the Banach space of all 2-component column vectors f(x) made up of continuous functions fs(x) and fc(x) defined on E, both tending uniformly to 0 as I x 1+ co and with norm

lIpIlL = IIfs llm+ llfc Ilm< aWe define the operator K(h) by:

Imh 20,

x2 # PL,,

(3.10)

fEB.

The Green’s function gU will must frequently occur in the combination K(A), and so we will concentrate here on establishing the properties in the Banach space B of the integral operator K(A).

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LEMMA 3.5. For fixed X in Im h > 0, X2 # pla , the operator K(A) is a bounded linear operator on B to B. Furthermore, (K(A)/)(x) is in B and Holder continuous even if/ is only bounded. Proof. Let II+ IIB = 1. Ikebe [8] has shown that for V E L, , V(x) = O(l x l-“-3 for j x 1+ co with E > 0 and llfc Ilm < 1, s G,(x, y, X) V(y)&(y) dy is Holder continuous and satisfies:

I/1,G&G

Y, 8

VYMY)

&

(1 -C cc

con&

(3.11a)

We must establish the corresponding properties for s G,(x, y, h) V(y)f,(y) dy. Since YE L,(E), the Holder continuity is just the assumption following Eq. (2.1Oa) on H, ; boundedness follows directly from the Schwarz inequality and the strong Carleman condition (2.1 la); and the integral goes to zero for large I x I by Eq. (2.1 lb):

)I/, Gdx,Y,4 V(Y)~,(Y)dv// < con% (3.11b)

j j-BGch Y,A)UY)A(Y)dy ;G

0.

Thus K(h) + E B and we have only used the boundedness of p

Q.E.D.

LEMMA 3.6. Forjxed X in Im h 2 0, A2 # p,, , the operator K(h) is a completely continuous operator in B. Proof. It suffices to show (see for example Hunziker for S = 1~~ I yn = K@) fn , II fn I/B < 11 the yn’s are

[12] or Regge [5]) that

(a) equally bounded (i.e., uniformly with respect to n); (b) uniformly equicontinuous; (c) maximized by M(x) for which lim,,l,, M(x) = 0. And these properties follow immediately from the proof of Lemma 3.5.

Q.E.D.

The distorted plane waves will be solutions to integral equations of the type f=y+K@)f. In the study of these integral equations the Fredholm

fundamental

importance:

(3.12) alternative [lo] will be of

POTENTIAL

SCATTERING BETWEEN CHANNELS

17

THEOREM 3.1. If K(h) is completely continuous in B, then the Fredholm alternative holds, i.e.,

either A(h) = [I - K(X)]-l -exists and iI A iI8 < co or the homogeneous equation f = K(h)f has nontrivial solutions in B. In addition to the properties of K(X) for fixed A, we will need to know something about the regularity of K(A) as a function of X. LEMMA 3.7. K(h) is continuous in X for Im h > 0 except at h2 = t.~~; i.e., given any E > 0 there exists a 6 > 0 such that (1K(h) - K’(h)ljB < Efor 1h - h’ j < 6.

Proof.

uw)~)(~) =

Gs(x,Y, 4 W)fX.d dy E G&GY, 4 W).L(y) dy i sE i* 1

The continuity of the scattering channel element of K(X) j in Im h > 0 has been established by Povsner [I I]. For the confined channel part take IIfS jjlo < 1 and use Eqs. (2.10) and the strong Carleman condition to obtain

IKGV -

w’))/lcw

= 1w - A21 ; (X2_ p5;;!2 _ & (?h3m j = 1(A” - X2)J‘,G,(x,Y,4 x [J; GAY, 2, A’>* W)f&)

dz] &I

< I Xf2 - A2 I . Etp II G,(x, .. W” j” I Uy)l dy

(3.13)

for I A’ - h / + 0 provided X2 # ,+ . Thus K(X) is continuous in both channels, and the continuity is uniform in any compact domain in Im h > 0 which does bot include X2 = pn . Q.E.D. LEMMA 3.8. For h2 # pn and x2 4 spectrum of K(x) the operator A(X) = [I - K(h)]-l exists and depends continuozrsly on X in the sense of the norm I/ !IB. LEMMA 3.9. Let D be a compact domain of the upper half h-plane such that D does not contain any eigenvalues of II, or the spectrum of K(h). Letp be the unique solution of the equation (3.14) lp^ = g” + K(3/A,

where#

E B is strongly continuous in h E D. Then,P is strongly continuous in X E D.

18

DASHEN, HEALY AND MUZINICH

The proofs of Lemmas 3.8 and 3.9 from Lemma 3.7 are identical to Ikebe’s proofs of the corresponding statements for ordinary potential scattering [8]. LEMMA

3.10.

K(A) is analytic in h for Im h > 0.

Proof. K(h) is analytic in h if [K(h)+’ Is and [K(h)p 1, are separately analytic in X for f (x) E B. The scattering channel element

is analytic in h by the usual argument [5, 121: for any simple closed contour C in the upper half h plane $ dx j dy Gk C

E

Y, 4 VY)~~(Y) = j dv f Gk E

c

Y, 8 VY)~,(Y) dh

=o

since the interchange of orders of integration is justified by the absolute convergence of JE G,Vfc dy and G,(x, y, A) is analytic in X for x # y. For the confmed channel we have:

where the interchange of summation and integration convergence of the sum

is justified by the absolute

= II VjCII . IIG,(x, *, 411 < const . II VI/ * II G&G e9@I.

Q.E.D.

LEMMA 3.11. The operator A(h) = [I - K(h)]-l is analytic in h in a neighborhood around any point h, for which A&) exists if K(h) is analytic around A,, .

Proof.

See Hunziker [12] or Regge [5].

We will also need some estimates of the asymptotic behavior of SEG,(x, y, A) x t(y) dy when we know something about the asymptotic behavior e(x).

POTENTIAL

SCATTERING

BETWEEN

19

CHANNELS

LEMMA 3.12. Let V(x) E L,(E). V(x) = O(/ x j-5/2-E)far 1x 1--+ 00 with E > 0, and g(x) E L,(E). Then for real X = a and 1 x 1 -+ co we haoe:

&al=1 2771x1

I E eciaE’gV(y) g(y) dy + O(i x /-3/2--F/2),

(3.15)

with i = x/l x j.

Proof. Let RI be fixed such that j V(x)1 d C I x I-5/2-E for 1x ( 2 R, and let I x I be so large that I x 11/2 = R > RI . Then separatef(x) into two pieces: fcx) = - &

[j-,,,<,

,;:-;,

J’-(v)&‘)

dy + /,Y,>R ly:-;

I V(y)g(y)dy]

= z + I’, where

dy

I’ < constIS

ld>R

1 x -Y

00

< const

r2dr

[S 141/2

F=

I2

V’(Y)

1, I gW2 4f]lp

112 l dcos 6 s-l (I x I2 + r2 - 2 I x I r cos 8)

1

Now we must estimate Z, which gives the leading term. For 1y / < R we can use Ikebe’s estimates [g]: eialz-vl = eiajzl 1 _ y [ 1 Ix-y/

and

+ Tl]5

1 =m

[ I+

y

+

11213

171 =

v2=o

q-g,;

y2. ( x2 1 '

20

DASHEN,

Imitating Ikebe’s treatment into five parts: eialzl

HEALY

sE e++~Vy)

-24x, m

g(Y)

27T1 x

1 s ivl
1

--

dy

+ &

J;Y,,R e-@g V(y) g(y) dy

JY,,R e-ia2’U(l - eio’z’nl) V(y) g(y( dy

V(Y)du

1

--

MUZINICH

[7] of the case c(x) = [V(x) g(x)] E L, we divide Z

+@aI4

I=

AND

477 1 x 1' s Id
2771x1

r12ei+-ffl

g(y)

B -yei+-YIV(y)g(y)dy

sE

e-iai’yV(y) g(y) dy + i Zj . j=l

Then we can put bounds on each Zj :

Finally we collect all the terms, and dropping the nonleading ones we have:

f(x)= -& jE,::-;l,

UY)dY)dY

eiQlzl e-iaJ1ey

=-2TlxI

sE

+

/--3/2--r/2)

0(1x

+

V( y) g(y) O(l

dy

x I-“).

Q.E.D.

LEMMA 3.13. Let V(x) E L,(E), 1 V(x)1 = O(l x l-3-G)for I x I -+ COwith E > 0, and let g(x) be bounded. Then for real h = a the function f(x) = SE G,(x, y, a) x V(y) g(y) dy satisjies Eq. (3.15) and the radiation condition:

,$ym I x I [@f/a I x I) - iaf Proof.

See Povzner [12] and Ikebe [8].

1= 0.

(3.16)

POTENTIAL

SCATTERING

BETWEEN

CHANNELS

21

The kernel G&c, y, A) has poles at the eigenvalues A2 = pn of H, , and so integral equation methods based on the integral kernel SU(x, y, A) will not be suitable for studying solutions of the Schrijdinger equation (1.11) in neighborhoods of these poles. However, the precise locations of the poles of G, are clearly irrelevant to the solutions to Eq. (1.11) since we could always choose a different separation of &’ into unperturbed Hamiltonian L%$’ and perturbing potential V’ in such a way that the spectrum of the confined channel part, H,‘, of PU’ does not contain a particular eigenvalue of H, and V’ contains a diagonal confined channel potential. The next two lemmas will make this statement more precise and provide a method for finding integral equations which are suitable for studying solutions of the Schrbdinger equation (1.11) in neighborhoods of eigenvalues of H, . LEMMA 3.14. Let the operator H, have eigenvalues (~~1. Then given aparticzdar eigenvalue pn , we can choose a boundedfunction U(x) of compact support such that for some real cy.> 0 the spectrum of the perturbed operator H,’ = H, + olU does not contain pn .

Proof. Choose U(X) positive semidefinite. For small (Y > 0 we can use lowest order perturbation theory to obtain the positions of the perturbed eigenvalues. Consider in particular the nth eigenvalue. The (possibly) degenerate eigenvectors will have their degeneracy split by the perturbation and the shift in the eigenvalue for each state is given (for small enough a) by:

dpn w const * 01sE5nW u(x) 4n(4 dx > 0. By suitably choosing U and taking 01small enough we can obviously arrange that the spectrum of H,’ does not contain the particular eigenvalue pcLra . Q.E.D. This leads us to consider the operator (3.17a) where

‘?“(x,Y,‘1 = (G”‘“by,‘) GJxyy,A))? UY) = WY>- *(Y) = %J) - aw9 (8

(3.17c)

and G,‘(x, y, A) is the kernel of the resolvent operator R,‘(h) = (h2 - HC’)-I.

(3.17d)

22

DASHEN,

HEALY

AND

MUZINICH

The new confined channel Green’s function G,‘(x, y, A) obviously has the same properties, as does G&c, y, A), except of course that its poles are slightly shifted. We must establish the relation between the solutions of the integral equations

fdx)= ,&a + J,~u(&Y9a v Y))4(Y) dY,

(3.18a) (3.18b)

when the inhomogeneous

terms are related through the integral equation,

92(x) = ,&)

+ s, ~u’(x9 YP4 WY) ,gl(Y) dY*

(3.19)

LEMMA 3.15. For A26 the spectrum of H, , H,‘, K(A), or K’(h) there is a single unique soZutiontt;(x) = fS(x) E B to Eqs. (3.18) and (3.19). In particular, for gl(x) = c&3 = g(x)@ th ere is only one solution fl(x) = fS(x) to Eqs. (3.18). Proof. Since A2 is not in the spectrum of any of the operators, each equation, (3.18a), (3.18b), and (3.19), has a unique solution (Theorem 3.1). Furthermore we have the relation between the Green’s functions

9t&‘(x,y, 4 = ~u(x, Y, 8 + 1 ~u’(& z, 4 @‘(d9.(z, Y, 4 &. We will show directly that fI is the unique solution to Eq. (3.18b) for f2 : ,92(x) + s, ~u’k = &)

Ys 8 WY) fdY> dY

+ s, ~u(x, YP4 ~(Y)fl(Y)

+ s, [~w’(& Y, 4 -

dY +

~u(x, Y, a1 VV)fi(A

- 9d-a

rgzcd

4

(3.20)

POTENTIAL

SCATTERING

BETWEEN

CHANNELS

23

Sincegd4 = g2(4 = &X.3 is a trivial solution to Eq. (3.19), !I = +‘z for this case follows immediately.

Q.E.D.

We have already mentioned in Section 2 that when we investigate the structure of the positive energy spectrum of the Hamiltonian X we will make the additional assumption that the off-diagonal potential V(x) is finite range. For that purposewe introduce the operator K,(h), defined on functions f (x) E (J?&!Z))~by: (3.21a) where Ydx,

Y, 8 = g&5

= e”‘“‘+W> = Vdv)

vdy)

(3.21b)

y, 4 f+“, (y

A).

(3.21~)

LEMMA 3.16. For Im X > --01 and A2 # p-L,, the Green’s function 9&q y, A) satisfies the strong Carleman condition. For V,(x) E L, (that is, when V(x) is$nite range) the operator K,(h) is a completely continuous operator of Hilbert-Schmidt type, and K,(A) is an analytic operator valued function of X in Im A > -CY with poles at A2 = pu, . The residue of each pole is an integral kernel offinite rank.

ProoJ Since e-++l is bounded, it is obvious that G&x, y, A) = G,(x, y, A) e-crlvi satisfies the strong Carleman condition for A2 # p., . That G&x, y, A) = GJx, y, A) e-@1 satisfies the strong Carleman condition for Im X > --01 is well known. Thus 9%,(x, y, A) also satisfies the strong Carleman condition, and it follows immediately that for V, E L,(E) the operator &(A) is Hilbert-Schmidt, and hence completely continuous in (L2(E))2. The analyticity of K,(h) can be proved in the same way as the analyticity of K(h) was proved in Lemma 3.10. The residue of the pole of (K,&)))(X) at then nth eigenvalue of HO is

where the sum is over the (finite number of) degenerate orthonormal corresponding to the eigenvalue CL,,of H, .

eigenvectors Q.E.D.

If we multiply both sides of the integral equation, (3.12), by V,(x), then we obtain fi = ~a + &@)pb, ,

(3.22a)

{a = Kf

(3.22b)

ga = v&J

(3.22~)

where for f, g E B, and are in the Hilbert space (L,(E))2.

24

DASHEN,

HEALY

AND

MUZlNICH

LEMMA 3.17. Let 9 E B. Then for every solution f E B to Eq. (3.12), fa E (L,(E))2 is a solution to Eq. (3.22a). And conversely iffy E (L2(E))2, then for every solution fa E (L2(E))2 to Eq. (3.22a)

+ (4 = ,&) + J; ~uar(x, Y, 4

+d Y) dy

is a soIution to Eq. (3.12) in the Banach space B. Proof. The first statement is obvious. And the converse follows immediately from the fact that 9&x, y, X) satisfies the strong Carleman condition (Lemma 3.16) Q.E.D. and pbrE (L2(E))2. Our principle tool in analyzing the structure of the positive energy spectrum of X will then be the analytic Fredholm theorem [13, 61. THEOREM 3.2. Let K,(h) be a compact operator-valued function meromorphic in some connected domain D of the complex A-plane, and let the residue of each pole of K,(X) be an integral kernel ofjinite rank. If [I - K&)1-l exists at some point X E D, then [Z - K&I)]-l exists and is meromorphic in D. The poles of [Z - K,(h)]-l occur at the discrete set of points {Xi> where the homogeneous equation fa = K,(h) pm has nontrivial solutions; and the residue of each pole is an integral kernel of$nite rank.

Proof.

See, for example, Tiktopoulos

[13].

When we investigate the properties of the resolvent W(h) = (h2 - Z)-’ as an integral operator in Section 6, we will want to use integral equation methods. The integral equation of interest will be just Eq. (3.14) with the particular choice of inhomogeneous term 8” = 9,(x, y, h); but ‘9% is not in our Banach space B and is singular at x = y. Nonetheless by iterating the integral equation, we will be able to write the kernel 9(x, y, A) of W(h) as a sum of terms with known singularities plus a bounded solution to an integral equation of the type (3.14) with g,, E B. With this in mind let us introduce the iterated kernels:

9)(x, y, A) = j SJX, z, A) q-7) =f+yz, Y, 4,

(3.23a)

~(“‘(x, y, A) = gtdx, Y,

(3.23b)

A).

We establish some regularity properties of these kernels by generalizing Ikebe’s analysis [8] of the corresponding kernels in the single channel scattering case.

POTENTIAL LEMMA

3.18.

SCATTERING

BETWEEN

25

CHANNELS

For Im h > 0, X2 # tag we have I JYX,

Y, 0

(3.24a)

< CC/l x - Y I19 1,

I ~Yx, Y,81 9 C(lnI x - Y l/lnP + I x - Al>4

(3.24b)

I ~Yx,

(3.24~)

Y, Ql d c * 4 S4)(-, y, A) E B,

for each jixed y,

(3.24d)

where the constants C are independent of x, y and are bounded in any compact domain in the upper half h-plane. Proof.

Write out P)

J(O)(x, Y, 4 = (

G,(x, z, A) 0 0

Jyx,

in more detail:

y, A) =

s Gdx,

0 Gdx, Y, 4 ) ’ z, 4 W

Gc(x, z, 4 W i I

F)(x,

G,(z>

Y, 8 dz

Gsk Y, 4 dz

0

=

0 zyx, y, A) ( zyy, x,X) 0 1’

Gs(x, z, A) V(z) GG(z,z’, h) V(z’) G,(z’, y, A) dz dz’ J’s

y, A) =

0

0 j-1 GA x, z, A) V(z) G,(z, z’, A) V(z’) G,(z’, y, A) dz dz’

I z12kYT 4 (

Jyx,

y, A) =

0

0

0 mx,

Y, 4 1’

Z(“(x, z, A) V(z) Z”‘(z, y, A) dz

s

Z’l’(z, x, A) V(z) Z’l’( y, z, A) dz

0

0

=

(Z@)(y, x, A) Z'YX, 0y, 4 1.

We know that for Im h > 0 the free particle Green’s function satisfies

and by the assumption Eq. (2.12)

I Gdx, Y, h)l < C/l x - Y I,

26

DASHEN,

HBALY

AND

MUZINICH

where C is bounded in any compact domain in Im h > 0 which does not include any eigenvalues of H, . Thus we can refer to Ikebe [8] for bounds on P, i = 1,2,3. P’ < C/l x - y /l/2, C “” ’ i C 1In 1x - y / I,

for for

I x - y j > l/2, I x - y I < l/2,

G C(ln I x - Y l/M2 + I x - Y II>, 1’3’ < c. For any compact domain in Im h > 0, A2 # CL,,, the constants C are bounded, and this establishes the bounds (3.24a, b, c), Then using Eq. (3.23a) and Lemma 3.5 we obtain N4)(., y, A) E B for fixed y. Q.E.D. LEMMA 3.19. condition :

For Im h > 0 the iterated kernels satisfy the strong Carleman II JY*,

Y, 411 G c

i =O, 1,2,3,4,

(3.25)

where C depends on A but not on y. ProoJ: we have

Since g&c, y, A) satisfies the strong Carleman condition for Im h > 0, II ~“(., Y, alI d c,

(3.26)

independent of y. We proceed by induction. Using the strong Carleman condition on gl((x, y, A) we have: jj P+l)(-,

y, X)/l” = SE) sESu(x, z, A) V-(z) Yi)(z, y, A) dz j2 dx < Sup II Gtd., z, h)l12[IE I +‘Wl ZEE

< c ’ II VII2 * II -T*,

* I XYz’,

y, A)/ dz]’

Y, 4112,

and with Eq. (3.26) this proves the Lemma.

Q.E.D.

LEMMA 3.20. For Im h > 0, h2 # p,, , and x # y the iteratedkernels f(i)(x, y, A), (i = 0, 1,2, 3,4) are continuous in x.

Proof. We know that NO)(x, y, A) is continuous except at x = y. To prove the assertion for i = 1,2, 3,4 we divide P+l) into two parts:

Jci+% YP4 = JreY,<6 cifu(x, z, A) V-(z) P)(z, y, A) dz

+L>d??“,(x,z, A) V(z) P)(z,

y, A) dz.

(3.27)

POTENTIAL

SCATTERING BETWEEN CHANNELS

27

By Lemma (3.18) each P)(z, y, A) is bounded in z for 1z - y 1 > 6 > 0. Thus the second integral in Eq. (3.27) is continuous (in fact, Holder continuous) by the same argument as in the proof of Lemma 3.5. In the first integral assume for the moment that / x - y I > S > 0. Then, G,(x, z, h) is continuous in x for all z such that / z - y / -C 8, and since V(z) and P)(z, y, X) are square integrable in z (Lemma 3.19) the first integral in Eq. (3.27) is also continuous in x for 1x - y / > 8. Finally since 6 can be made arbitrarily small, we have proved that Wi+l)(x, y, X) is continuous in x for x # y. Q.E.D. Thus we see that for fixed hz # pn the iterated kernels P)(x, y, h) are no more singular in x, y E E x E than are the iterates of G,(x, y, h) V(y) alone [8]. 4. ASYMPTOTIC

COMPLETENESS AND UNITARITY

OF THE S-MATRIX

First we will establish the self-adjointness of X = tiU + V. This is a trivial consequence of the Kato-Rellich theorem: THEOREM 4.1. Let &$ , dejined on the domain 9(Xu), be self-adjoint and bounded from below; and let V be relatively bounded with respect to 3Eb, with Zu-bound less than 1. Then %? = Yu + V is self-adjoint and lower semibounded on the domain 9(H) = B(ti,).

The proof is given by Kato [7]. The relative boundedness of V was proven in Lemma 3.2, so the conditions of the theorem are satisfied. Let X’F and .%?‘a~be the spectrally absolutely continuous parts of YU and &‘, and let PUc and 98~ be the orthogonal projections on the spaces ktc and &C of absolute continuity of tiU and z?, respectively. Obviously in the case of interest g’“,” is the projection on the scattering channel: (4.1) As usual in scattering theory we introduce the one-parameter family of unitary operators w(t) z &fte-i*ut, -co
tiu) = S-Jim

W(t) 8, .

(4.3)

These wave operators (if they exist) take free scattering channel states into inter-

28

DASHEN,

HEALY

AND

MUZINICH

a&g states which look like free states in the asymptotic past (IV-) or future (IV+). If we refer to the ranges of W- and W+ as Ain and Lout, respectively, then:

The S-matrix should take free scattering channel states into free scattering channel states and it should have no confined channel matrix elements. Thus it is natural to make the definition [4, 6, 71: s = w++w-.

(4.4)

We expect S to be a unitary operator with respect to the scattering channel. However, this is not obviously true since it requires: Ain

=

&out

(4.5)

(Simon [6] calls this “weak asymptotic completeness”). In fact Kato and Kuroda [14] have constructed an example in ordinary potential scattering of a non-local potential in E2 for which Ain # AOut. It is worth noting that with respect to the scattering channel our problem is effectively a non-local potential problem. Nonetheless we are able to prove a stronger form of asymptotic completeness than Eq. (4.5) by a direct application of a powerful theorem on generalized wave operators [7, 8, 151: THEOREM 4.2. Let PU be self-aa’joint and bounded from below; and Iet Y” be relatively bounded with respect to HU with %$-bound less than 1. If V can be written as a product of two factors, V = VT+““, such, that both V’(&$ - y)-’ and V”(T& - y)-’ belong to the Schmidt classfor some y smaller than the lower bound bound of tiU , then the generalized wave operators W.&F, XU), exist and are complete and the S-matrix is unitary.

The proof is given by Kato[7]. Referring back to Lemmas 3.2 and 3.4 we see that the conditions of the theorem are satisfied, and that the only restriction on V needed is V E L, n L,(E). The particular form of asymptotic completeness established in Theorem 4.2 is referred to by Simon [6] as “Kato asymptotic completeness.” This means that W* are p.artially isometric operators with initial set &r and final set Ain = AoUt = A? w+tw*

= P’“,“,

w*w*+

= PSC,

w* = w&y

= LPc w, )

(4.6)

POTENTIAL

SCATTERING

BETWEEN

CHANNELS

29

and the operators Z”,” and #a~ are unitarily equivalent: SW& = W&scu.

(4.7)

We will use this “Kato asymptotic completeness” in our proof in Section 6 of the eigenfunction expansion associated with %. Then we will use the eigenfunction expansion to prove a stronger form of asymptotic completeness, which Simon [6] calls “asymptotic completeness;” the particular property which will be added to those already proved will be A”c = (Abound)l. Simon [6] identifies a more restrictive form of asymptotic completeness, which he calls “strong asymptotic completeness,“by the additional property that all bound states have nonpositive energy. However for a specific class of Hamiltonians .Z which satisfy all of our requirements we have explicitly constructed positive energy bound-state solutions [9]. Thus we cannot hope to prove “strong asymptotic completeness” in general for the operators Z. In Section 8 it will be useful to have a representation of matrix elements of the wave operators W* . To establish this representation we will frrst need some results on various limits. Our proofs will be patterned after Simon’s [6], but of course our Hamiltonians are not defined as quadratic forms. THEOREM

4.3.

If!, JZE 9(X)

= 9(3$),

then

(4.8a) and 1jq-jW>(ya, (eist Proof.

l>f> = j@, sy3.

(4.8b)

Working in a spectral representation for 0 = X or J& we have 9 = g(k),

f = f’ 6’4

and (l/t)s+(k)(eiot

- l)!(k)

= (l/t)s+(k)(eiKzt - l)f(k)

converges pointwise to ig+(k) k”f (k). Then since (l/r) 1eifizt - 1 / < / k I2

and

(g+(k) I k l”JGW 6 LI

for 8, PE g(O), Eqs. (4.8) follow from the dominated LEMMA

4.1.

Let

convergence theorem. Q.E.D.

f (t) be a bounded measurable function and let l,@ ltf(t’)

+, ”

dt’ = a.

(4.9a)

30

DASHEN,

HEALY

AND

MUZINICH

Then !J’ i 0m

e-ctj(t) dt = a.

(4.9b)

The proof is given by Simon [6]. By combining Theorem 4.3 with Lemma 4.1 and the definitions (Eq. (4.3)) of W* we get the desired expression for the matrix elements of the wave operators: THEOREM

4.4.

Iff,

9 E .9(X)

= .C&%$), then:

(9, (W+ - PJ f) = l$r Joim i(p, eiar”tVemixutB,f) erctdt. Proof. Noting Theorem 4.3 :

that e- iJ$utpRE g(&)

(d/&)(8,

ei*teeixutP8f)

(4.10)

and e-b*tg E C#(.%‘) we have from = i(g, ei~t~e-idPutB,

(4.11)

f ),

and from the definition (Eq. (4.3)) of WA ,

(9, (w+ - @J f) = lipm lt i(y, eipt’Ve-iHut’8,f)

dt’.

(4.12)

Using the fact that e- i-@utf E 5@(Xu) is bounded (Lemma 3.2) and the Schwarz inequality we easily see that the integrand is bounded I@, eimtVeCi”“t8,f)12

< [I ewimt9 11- 11Ve-idE”,tiP,fll

G c . II9 II * II VII d c

(4.13)

since we always assume that V E L,(E). Then Eq. (4.10) follows from Lemma 4.1. Q.E.D.

5.

THE

SPECTRUM

OF X

We expect that the spectrum of &’ will consist of a discrete set of normalizable negative energy bound states, a continuum of positive energy scattering states, and possibly some discrete bound states embedded in the continuum. Since &’ is self-adjoint we know of course that its spectrum is confined to the real energy axis. The negative energy spectrum is easiest to deal with.

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LEMMA 5.1. On the negative energy axis S has no continuous spectrum. If there are negative energy eigenvalues they are discrete and of finite multiplicity; they form an isolated set with the origin as the only possible limit point.

The proof is given by Ikebe [8] and relies on the facts that # and ZU are lower semibounded and for real a > 0 the operator V(XU + a)-l is Hilbert-Schmidt (this follows directly from V E L, and the strong Carleman property of gU). To examine the bound-state spectrum of J? in detail we will use the integral equation (3.12) and the Fredholm theorems (3.1) and (3.2). Of course the operator K(X) does not exist at the discrete set of points X2 = p”n on the real energy axis, but Lemmas 3.14 and 3.15 tell us how to handle this difficulty: Throughout this section we can shift the poles of K(A) by making the replacements:

a4 + Jm, Y-(x) + v-(x) - G?(x)

(5.1)

All of our results go through equally well with this substitution, so the restriction h2 f pn which is implicit in the following is an artificial one. The correct interpretation of each result requires that for given A we always choose a K(X) which exists at that point. Following Ikebe [8] let us establish a correspondence between bound-state solutions to the SchrSdinger equation (1.11) and solutions to the homogeneous integral equation @ = K(h) @.

(5.2)

LEMMA 5.2. For Im h > 0 the homogeneous integral equation (5.2) has nontrivial solutions in B tfand only ifA is an eigenvalue of Z’. In particular for Im X2 # 0 there are no nontrivial solutions to Eq. (5.2).

Proof: V@ is square integrable (since V E L,(E) and @ E B), so V@ is contained in the domain of W,(X). Thus we can write Eq. (4.2) as

(5.3) Operating on both sides of Eq. (5.3) with (tiU - P) we find: (2% - h2) CD= --v-CD,

or

32

DASHEN,

HEALY

AND

MUZINICH

From YE L,(E) and the strong Carleman condition on 9, it is easy to show that @ E (L,(E))2: (@, @> = 1, j j-, gu(x, Y, 4 V(Y) Q(Y) 4 I2 dx (5.4)

where we have used the Schwarz inequality and Fubini’s theorem. eigenvector of .%’ with eigenvalue h2, and since X is self-adjoint there are no nontrivial eigenvectors for Im A2 # 0. Now let X2 be an eigenvalue of 9P with eigenvector @ E 9(Z) know by Lemma 3.2 that @p(x)is a bounded function, and so V@ is contained in the domain of W,(A). Thus we obtain

It follows from Lemma 3.5 that @ E B, which completes the proof.

Thus @ is an (Theorem 4.1) = 9?(HU). We E (L2(E))2 and

Q.E.D.

We must also establish this correspondence for positive energy eigenfunctions: LEMMA 5.3. For real positive h = a in the spectrum of K(h), let @J(X)E B be a solution to the homogeneous integral equation (5.2). Then Q(x) is a solution to the Schriidinger equation (1.11) and O(x) E (L2(E))2.

Proof. We first observe that Q,(x) E L,(E) follows from Eq. (5.5) and the strong Carleman condition on G, for a2 # p., :

where we have used V E h(E), theorem. Then as is given by

0 E B, the Schwarz inequality,

@dx> =sGk

Y, 4 V(Y) @du> dv

and Fubini’s

(5.7)

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and by Lemma 3.12 we have: G,(x) = --(ei~l~l/2n 1x I) j eciaL.yV(y) Q,(y) + O(l x 1--(3/2)+/2))

(5.8a)

for 1x1--+ co. Thus we will have QjsE L,(E) provided the integral

fs = j

e-iai’y V( y) dsc(y) dy

(5.8b)

vanishes. To show thatf, does vanish we operate on Q(x) with (XU - uz), and use Green’s formula. For a2 f TV, we have V@, E L,(E) so V@, is contained in the domain of R,(u) and:

(Ho-u2)@,= -ms.

(5.9a)

Since a2 is in the continuous spectrum of H, , the operator T,(a) defined by Eq. (3.19) is an extension of R,(a), and we cannot conclude directly that (H, - uz)[Ts(u) J4DJ = - IQ, . However, Q. is Hijlder continuous (by Lemma 3.5) and T/ has been assumed to be HGlder continuous, except at a finite number of points; thus V@, is Holder continuous, and we can apply the formal differential operator (-Vz - u2) to obtain: Ws - a”) Q,(x) = (-v2

- a”)(--l/277) jE (eialr-lJI /I x - Y I> V(Y) @c(Y) dY

= - c-4 @c(x),

(5.9b)

for x different from the (finite number of) singularities of V(x). Combining Eqs. (5.9) we have: (Zu - a”) a(x) = --r@(x), (5.10) for x different from the singularities of V(x). Now using Eq. (5.9b) and Green’s formula we find:

jKfR)[@c+(x)WI @s(x)- @s+(x)WI QicWl dx = jKtn) N@,‘(x)Hs@sW

- (@s+(x) Hs@dd>+l dx =s S(R) Ws’MW)

@s(x)> - (@sWWW@s(x))+1 4

(5.11)

34

DASHEN,

HEALY

AND

MUZINICH

where K(R) is a sphere of radius R centered at the origin, S(R) is the surface of K(R), and n is the outward normal. (For a discussion of the validity of this equation when V has a mute number of singularities see Ikebe [8].) Let us first estimate the left-hand side of (5.11) for large R.

where we have used. V(x) E L,(E), V(x) = O(l x le6iSe)for large I x [, and the strong Carleman condition on G, . Thus the left-hand side of Eq. (5.11) can be made arbitrariily small by choosing R sufficiently large. The surface integral on the righthand side of Eq. (5.11) can be well approximated for large R by substituting for a,(x) its asymptotic behavior, Eqs. (5.8). Dropping terms in Eq. (5.11) which vanish in the limit R -+ 00 we find: 2a S(R)I fa I2 dQ = 0, s

(5.12a)

and since )fa Jais positive semidefinite, we must have

fs = s, Y$(Y> W Y(Y) Q’(Y) 4 = 0, which completes the proof.

(5.12b) Q.E.D.

We should point out that if there is a zero energy solution to Eq. (5.5), @ = K(0) @, it need not be square integrable since in that case Eq. (5.12) becomes an identity and @, is not necessarily in L,(E). Of course this is nothing new since the same problem can odcur in ordinary potential scattering. LEMMA 5.4. For real positive h = a the homogeneous integral equation (5.2) has nontrivial solution in B if and only if a” is an eigenvalue of S. And every eigenvector of .%’ with eigenvalue ha > 0 is a solution to Eq. (5.2).

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Proof. From the previous lemma any solution @ E B to Eq. (5.2) is a square integrable soution to

(5.13)

&@ = a2@,

which proves the necessity. To prove the sufficiency and the last statement, let @ be an eigenvector of .?‘Pwith eigenvalue a2 > 0. From Lemma 3.2 we know that @ E B, so V@ E (L,(E))2. Thus V@, is in the resolvent set of H, and

@,W= fEGchY,a>V(Y)@du) 6~.

(5.14)

Furthermore, it follows from Eq. (5.12) and Lemma 3.5 that qbC(x) is Holder continuous in X, so by the same argument as used in Lemma 5.3 we have: (5.15) the right-hand side is easily seen to exist since V E L,(E), 1x 1 -+ cc, and @, is bounded. If we define

V = O(l x /-5/2-6) for

then by Eqs. (5.13) and (5.14), 6, is an eigenvector of H, with eigenvalue a2, and it is easy to see (as in the proof of Lemma 5.3) that $‘, E L,(E). However, H, has no nontrivial eigenvectors in L.,(E), so sS = 0 and we have shown that

which completes the proof since Lemma 3.5 assures us that @ E B.

Q.E.D.

We have not yet pinned down the nature of the positive energy bound-state spectrum. To do this we make the stronger assumption that the potential V is &rite range, i.e., there is a positive constant a,, such that V,(X) zz e+l V(x) E L,(E)

Multiplying

for

u < 01~.

(5.18)

both sides of the integral equation

/cd = 544 + 1 ~u(x,Y,4 ~(‘-(y)p(Y)4

(5.19)

36

DASHEN,

HEALY

AND

MUZINICH

by Va(x) and defining

(5.20)

we obtain (5.21)

To each solution 4 (x) E B of Eq. (5.19) for Im h > 0 corresponds a solution f=(x) E (L,(E))2 of the new integral equation (5.21) (by Lemma 3.17); in particular to every solution of the homogeneous equation (5.22a) corresponds a solution of the homogeneous integral equation

Furthermore we know from Lemma 3.16 that K,(h) is a completely continuous operator of Hilbert-Schmidt type, and that it is an analytic operator valued function of X in Im h > --ol; it has simple poles at the eigenvalues pLnof H, and the residue of each of these poles is an integral operator of finite rank. It follows from the analytic Fredholm theorem (3.2) that the spectrum of K,(A) forms a discrete set, with only a possible accumulation point at X = + co. Since the spectrum of K(h) is also the spectrum of K,(h), the same statements apply to the spectrum of K(h). Combining this with Lemmas 5.1 through 5.4 we have: THEOREM 5.1. If V is finite range, then the spectrum {Em = hn2} of K(h) forms a discrete set on the real energy axis. The spectrum is bounded below, and the only possible accumulation point of eigenvalues is E = + co; each eigenvalue is ofjinite multiplicity. The homogeneous integral equation (5.2) has nontrivial solutions in B tf and only tf @j(x) = Y,(x) is an eigenvector of X corresponding to the eigenvalue A2 = E, (exceptperhapsfor a zero energy solution to eq. (5.2) which is not in (L,(E))3.

Proof It remains only to show that the multiplicity of each eigenvalue is finite; but this follows immediately from the fact that K,(A) is a completely continuous operator in the space (L,(E))2. Q.E.D.

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6. THE RESOLVENT OPERATOR W(X) = (X2 -ri">-" In proving the eigenfunction expansion associated with Z? in Section 8 the properties of the resolvent W(A) and its Green’s function 9(x, y, A) will be essential ingredients. We will establish the required properties in this section. THEOREM 6.1.

Let Im X > 0 and h2 # E, ; then

(a) B(A) is an integral operator of Carleman type and its kernel 9(x, y, A), called the resolvent kernel or Green’s function, satisfies the equation

9(x, Y, 4 = gu(x, Y, 3 + I, gu(x, z, 4 v(z) ‘%, Y, 3 dz

(6.1)

as a function of x a.e. in E for a.e. y E E (a.e. means almost everywhere or almost every).

(b) If ~F(x, y, A) satis-es the integral equation (6.1) and 9(-, y, A) E (L,(E))2 for eachfixed y, then %+T, y, A) is the resolvent kernel of 92(h): mx,

Y, 4 = e,

Y, 4

(6.2)

for a.e. x, y in E x E. Proof. The proof is almost a word for word transcription of Ikebe’s proof [8] of the corresponding statements for ordinary potential scattering with L, potentials; we mention that here the correct symmetry property of 9 is given in Lemma 6.1 below; and in the proof of (b) our bounded kernel 9@)(x, y, A) (see Lemma 3.18) replaces Ikebe’s A(3)(~, y, A). Only the neighborhoods of the points A2 = pfl require a little caution. In those neighborhoods we can replace the original integral equation (6.1) by 9(x, y, A) = 92c’(x, Y, 4 + 1 Su’(

x, z, Q[-tr(z)- %)I g(z,Y, 3 dz (6.3)

with Yu’ and (%Y given by Eqs. (3.17). Using the same argument as in the proof of Lemma (3.15) we see that the solution of Eq. (6.3) is also a solution of Eq. (6.1), provided A2is not in the spectrum of H, or H,‘, so by (b) of the theorem there is a single unique solution to both Eqs. (6.1) and (6.3) at any value of X outside the spectra of H, , H,‘. Of course the same arguments used to establish the uniqueness of the solution to Eq. (6.1) can also be used to establish the uniqueness of the solution to Eq. (6.3) at the points X2 = pn of the spectrum of H, where the integral equation (6.1) is ill de&red. Throughout this section we will take Eq. (6.1) to be symbolic in the sense that it really stands for Eq. (6.3) with @ chosen so that A2 is not an eigenvalue of H,‘. Q.E.D.

38

DASHEN, LEMMA

6.1.

HEALY

AND

MUZINICH

The resolvent kernel 9(x, y, A) is symmetric: w,

Y, 8 = ,mY,

(6.4)

-% 4

for a.e. x, y in E x E. Proof. The proof follows Ikebe [8 ] word for word with the obvious modtication Q.E.D. of the inner product required by the fact that our kernels are matrices.

The kernel equation (6.1) is not in the best form for rigorous investigation because the inhomogeneous term is singular at x = y. However Ikebe [8] has shown that this difficulty can be easily overcome by iterating Eq. (6.1). Successive iterations produce successively less singular inhomogeneous terms, and after four iterations we reach an integral equation with inhomogeneous term in our Banach space B. Then following Ikebe [8] we have: THEOREM 6.2. Let Im h > 0 and X2 # E,, , and let 9(*)(x, y, A) satisfy the mod&!ed kernel equation :

@4)(x, y, A) = 4(4)(x, y, A) + J, 3,(x, z, A) q4

@Yz, Y, a.

(6.5)

For eachfixed y there is a unique solution g(*)(*, y, A) E B of eq. (6.5), and

59(x,y, A) = i S3)(x, y, A) + @*Yx,Y,

4

(6.6)

&O

is, for eachfixed y, a continuous function of x for x # y. For Im X > 0 (and A2 # E,,) e,

for a.e. x, y E E

x

y, 4 = %(x9 Y, a

(6.7)

E, and G?satisfies the strong Carleman condition

II@c,Y, 4ll

< c

w3)

where C is independent of y. ProoJ: By Lemma 3.18 we have J(*)(*, y, A) E B for P # pn and each fixed y, so by Theorem 3.1 there is a unique solution @4)(*, y, A) E B to the modified kernel equation (6.5) for each tied y and A2 # p,, , he # E, . Lemma 3.18 also tells us that J(f)(x, y, A), i = 0, 1,2, 3, are continuous in x # y for A2 f’ pn , and it

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follows that 4(.x, y, A) is continuous in x # y. Since Ju)(x, y, h) i = 0, 1,2, 3,4 satisfy the strong Carleman condition (Lemma 3.19) and

I/jE~u(x,z,4 +w @4)(z, Y,a 11 S C*IIgu(-,z,4 s,I W’>ldz’- II@4Y., Y,QIB < c, the modified kernel @(x, y, h) must also satisfy the strong Carleman condition:

(6.9)

II%*, Y, 3l -==L c.

Then by iterating the resolvent equation (6.1) we see that 9 and 59 satisfy the same integral equation, and by Theorem 6.lb we have @(x, y, h) = 9(x y X) for a.e. x, y, E E x E, which is Eq. (6.7). Of course since B(x, y, h) is symmetric (Lemma 6. l), all the above statements apply equally well to 9(x, y, h) as a function of y for fixed x. As always, Eqs. (6.5) through (6.8) are symbolic in the sense specified in the proof of Theorem 6.1, and all restrictions to X2 f pn in the proof are artificial. Q.E.D. In Section 7 we will want to relate the distorted plane waves to the resolvent kernel. To do this we will treat the columns of @ separately. Let (6.1Oa)

The columns of @ satisfy the integral equations @)(x, y, A) = B:'(x,

y, A) + j, B,(x, z, 4 "tr(z> @')(z, Y, 4 dz

z.zzB~)(x, y, A) + jE @(x, z, A) v(z) @?(z, Y, 4 dz,

@)(x, y, A) = 4$)(x, y, A) + j, @u(x,z, A)%I

(6.114

@c'(z,Y, A>dz

= @:‘(x, y, A) + jE g(x, z, h) V(z) @‘(z, y, A) dz.

(6.1 lb)

40

DASHEN, HEALY AND MUZINICH

We defme the conjugate Fourier transform 2(x, k, A) of @8)(x, y, A) by: 2(x, k, A) = (2~r)-~/~/e @‘(‘)(x, y, A) eik’g dy

(6.12) = sE @(x, Y, 4 YT(Y, 4 dy, for Im h > 0, A2 # E, , and all x. We will see in Section 7 that the distorted plane waves are directly related to 2(x, k, A). Here we will establish one simple condition ong.

LEMMA6.2. If Im A > 0 and A2 # E, , then @‘@)(x,y, A) is absolutely integrable in x and in y, and 2(x, k, A) is a boundedfunction of x and k for x E E and k E M. For the proof see Ikebe [S].

7. EXISTENCE OF CONTINUUM EIGENFUNCTIONSAND THEIR RELATION TO THE RESOLVENTOPERATOR92(h) = (h2 --X)-l

By analogy with ordinary potential scattering theory we def?ne the distorted plane wave to be the bounded solution Y(x, k) to the SchrSdinger equation (7. la)

S’Y(x, k) = 1k I2 Y(x, k),

which satisfies the asymptotic condition Y.&G W = (‘WG k> - YAx, W, = 00 x I-‘),

I x I large,

(7Sb)

and the radiation condition

I x I W/d I x I>ydx, k) - ikydx, WI x

0.

(7.k)

In Eq. (7.lb) ‘u,(x, k) represents the free plane wave: ul,(x, k) = (2n)-3/2eik.z i . 0

(7.2)

Then Y,(x, k) has the form (plane wave) + (outgoing wave). The distorted plane wave will be shown to be the solution to the integral equation: (7.3)

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Of course this equation is symbolic in the sense that we must always choose a K(A) or K’(X) which exists at the point X = k; by Lemma 3.15 and Eq. (7.2) the distorted plane wave Y(x, k) does not depend on the choice of K(h) or K’(A). The free plane wave Y?(x, k) is bounded, but it is not in our Banach space B. If we define (7.4) @(x, 4 = Yx, 4 - Yk

k),

then we can rewrite the integral equation (7.3) as (7.5)

and by Lemma 3.5 Qf(x, k) E B. THEOREM 7.1. There exists a unique solution @(x, k) of the integral equation (7.5) for / k I > 0, k2 # E, ; and

(7.6a) is a solution of the integral equation (7.3) and of the Schriidinger equation (7.la) with the asymptotic condition (7.lb). Both Y(x, k) and @(x, k) are bounded and uniformly continuous in x and k for x E E and k E D, where D is any compact domain of A4 not containing any eigenvalues of 2’. If we define Y(x, k) for k2 = E, by Y(x, k) = ,!lrnE Y/(x, k’), -‘?I

(7.6b)

which is also a bounded solution to Eqs. (7.3) and (7. la) with the asymptotic condition Eq. (7. lb), then Y(x, k) is untformly continuous in any compact domain of M. If V(x) satisfies in addition the stronger condition V(x) = O(( x j-3-‘), 1x 1+ co with E > 0, then Y(x, k) satisfies the radiation condition (7.1~). Proof. Since cD~(x,k) E B, Theorem 3.1 tells us that Eq. (7.5) has a unique solution @(x, k) E B for k2 # E, . By Lemma 3.5 Qr(x, k) and @(x, k) are both Holder continuous in x, and using the same argument as in Lemma 5.3, Y(x, k) must satisfy the Schriidinger equation (7.la). To prove continuity in k we first show that

tDf(x, k) = (T’)-~/~s, G,(x, y, k) V(y) ei7<.Ydy (3

(7.7)

42

DASHEN,

HEALY

AND

MUZINICH

is strongly continuous in k E D: I(@,(x> k) - @Ax, Wh I

= (2~)+~ / lE [G,(x, y, k) eik*y - G&x, y, k’) eik”“l V(Y) dy 1 = (2~)-~‘~

) s, K&(x,Y,k) - Gc(x,Y,W eikWA 4~

+ s, G,(x, y, k’)[eik’Y - .F~~‘*~]V(y) dy 1 < c I kt2 - k2 I . / s, G&c, 2, k) (s, G&s Y, k’) e”“‘” W)

+ CJ;yl
I ‘Xx, Y, k’) JWl

4) dz 1

* I& - k’) * Y I 4

,Y,>R IG&Y, WI *IW)l dr +cs =

I1

+

12 +

(7.8)

43 ,

where we have used the first resolvent equation [16] [R,(A) - R,(X)] = (X2 - A”) R,(A) R,(X)

(7.9)

to obtain the expression for II . Then using the conditions (2.14) and (2.15) on V and the strong Carleman condition for G, we get: I1 < C - / k’2 - k2 I [Max Sup /I G,(x, *, 4111j.E I W)l A=k.k’

seE

&

< C * ) k’2 - k2 I, 12

< I k - k’ I . R * Sup II Gcb, -3VII . II VII XCZE

(7.10)


4, < C - Sup II G&x, -3WI1[J;,,,, y2W 4fi2 XEE

< CR-I- .

Choosing ] k - k’ ] small and R = Ik - k’ I-li2 we have: I1 + I2 + I3 < C j kf2 - k2 I + C I k’ - k j1i2.

. (7.11)

Thus @,(x, k) is strongly continuous (in fact, Holder continuous) uniformly k E D, and by Lemma (3.9) @(x, k) is uniformly continuous in k E D.

in

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To study Y(x, k) for k2 in the neighborhood of an eigenvalue E, of X, we use the Hilbert-space formulation equations (3.21) and (3.22). That solutions to Ydx, 4 = Y&,

(7.12a)

W + t&(k) ‘u,t*, k))(x)

exist for k2 = En follows [lo] from Eq. (5.12b), which guarantees that the solutions (Y:,)(x) = Y,+(x) e-“lzl E (L,(E))2 to the adjoint homogeneous equation WkJW

= emmiejE ~,A

Y, W ~&W’%9

(7.12b)

4

are orthogonal to the homogeneous term Yfa(x,k) in Eq. (7.12a). By the analytic Fredholm theorem (2.2), in a small enough neighborhood d of En we can write the resolvent operator 9&(h) = [Z - K,(h)]-l as ~3)

(7.13)

= T,/G2 - En) + WO

where S,(h) is analytic in h E d and T, is the orthogonal projection [lo, 131 onto the subspace spanned by the set u of orthonormal solutions Y,,, E (L2(E))2 to the homogeneous equation ~ndx) = WW

ul,JC4

Using Eq. (5.12b) and the analyticity O(l k 1 - E1lz), we see that

k2 = E,, .

(7.12~)

in k of JE Yn+(x3)V(x) Yf(x, k) dx =

G”-d4/@2 - Ed y,,(., 4)(x) = c W~XY@~ - En)) jE + = ; tYn&M~2

- En)) j Y&)’

y&, +w

A) dx

u,tx, 4 &

exists and is continuous at X2 = En . And since S,(X) is analytic in X E d, Fa@> Y,d*,al( x 1is certainly continuous (in the sense of the L, norm) at X2 = E,, : II %N y,cd*, A) - Wo)

Yd.7 u2

< IIWV - xx@o~l12 IIYd-, U12 + IIwv~12 . IIYd*, Q - Yd*, uz < c . 1h - A, 12. Thus yak

4 = W&V/O2

- En) + X44) yd-,

4)(x>

is continuous with respect to the L, norm, and using Lemma (3.17), Wx, 4 = %tx, W + jE ~ua(x, Y, W yatv, 4 &,

44

DASHEN,

HEALY

AND

MUZINICH

it follows from the continuity of S&C, y, k) and ?PU(y, k) that UC&, k) is continuous at k2 = E, . The boundedness and continuity in x of Y(x, k) for k2 = E, can be established by the same argument as used above for k2 # E, . The asymptotic condition (7.1 b) follows from Lemma (3.12), and if V(X) = O(l x /-3-E) for x large, then the radiation condition (7.1~) follows from Q.E.D. Lemma 3.13). We must point out that while Theorem 7.1 does not use directly the assumption that Vis finite range (except in defining Y(x, k) at k2 = E,), it would be an empty statement if the eigenvalues E,, formed a dense set on the real energy axis. We will always use Theorem 7.1 in conjunction with Theorem 5.1, which does explicitly require that V be linite range. We must now relate the distorted plane waves Y(x, k) to the Fourier transform 2(x, k, h) of the first column of @(x, y, A). As usual we model our treatment after that of Ikebe [8]. Let us detie (7.14)

&, k, A) = (A2- I k I3 ,&G k, 4.

LEMMA 7.1. For Im h > 0, A2 # E, , and x E E the function 8(x, k, A) satisjies the integral equation:

8(x, k, 4 = K(x, W + JE~u(x, z, 4 V4&, Proof. s, W,

k 4 dz.

(7.15)

From the kernel equation (6.1) we have Y, 4 %ft~)

dy = j-, gu(x, Y, 4 K+(Y) + s, ~tAx> ~4

W>

4 (s, ‘%> Y, A>y’,tpty) du) dz, (7.16)

where! (4 E (~5~69 2 and 8, is the projection operator Eq. (4.1) onto the scattering channel. Introducing the Fourier transforms j,(k) = (27~‘)-~/~I e--ikms(l,O)+(x) dx E

(7.17)

= I E Yf+(x, k) f’(x) dx, and

u,(x)m2 - 1k 12) = (21~)-~/~j, gu(x, y, A) (i) eik.g dy z-z

JE ~utx, Y, 4 %CY, 4 4,

(7.18)

POTENTIAZ, SCATTERING BETWEEN CHANNELS

45

and using Parseval’s equality we can put Eq. (7.16) in the form:

where the interchange of orders of integration Fubini’s theorem, since 1 jE gu(x, z, 4 v(z) ( jE +“(z, k, +&V

in the last term is justified by

dk) dz 1 G C . II ~u!x, -3h)ll . II V - 11?P(z 3 *>Qll * llrpll < c.

(7.20)

Noting that p (k) E L,(M) is arbitrary we obtain from Eq. (7.19)

f”(x, k, 4 = YAx, WW - I k I”>+ jE ~u(x, z, 4 +%>@, k 4 dz and multiplying

by (X2 - 1k I”) we have Eq. (7.16).

(7.21) Q.E.D.

LEMMA 7.2. The definition of 8(x, k, A) can be extended to Im h 3 0 for X2 # E, . 8(x, k, A) is bounded and un$ormly continuous in x E E, k E M, and Im X > 0, 0 < (II < Re A < /3 < co, provided no bound states of A@ are included in [ar, p]. In particular

g(x, k, I k I) = Y(x, k). Proof.

(7.22)

If we define XAX, k 4 = I E g&,

Y, 4 T(Y)

4 4s

Y,(Y,

(7.23)

and x(x, k 4 = &,

k 4 - Y&G W

(7.24)

then x(x, k, A) satisfies the integral equation x(x, k 4 = x&, k 8 + jE y”dx, Y, 4 T(Y)

X(Y, k 4 4.

(7.25)

The methods used to prove Theorem 7.1 can be applied in an obvious way to show that &x, k, A) E B is strongly continuous uniformly in k E A4 and A for Im h > 0 and 1 A2 - E, I > E > 0 for all n. By Theorem 3.1 there is a unique solution x E B to Eq. (7.25) for Im A > 0, A2 # E, ; and this solution is given by Eq. (7.24)

46

DASHEN,

HEALY

AND

MUZINICH

since X(X, k, X) = b(x, k, A) - Y,(x, k)) is b ounded (by Lemma 6.2) and in our Banach space B (by Lemma 3.5 and Eq. 7.25). Now write Eq. 7.25 as x(x, k 4 = V - GW1

X&G k, 8.

(7.26) Using the strong continuity of xB established above and Lemma 3.9 we deduce that x(x, k, A) is strongly continuous in k E M and A for Im X 2 0, 1 X2 - E, 1 > E > 0. Thus the definition of x(x, k, A) (and hence 9(x, k, A)) can be extended to Im h > 0, A2 # E, and the required continuity properties of 9:(x, k, A) follow immediately. In particular since XAX, k I k I) = Y,(x, k), for X = [ k 1the function 8(x, k, A) is just the distorted plane wave:

(7.27)

(7.28) ,&, k, I k I) = u,(x, W. Needless to say it is implicit in our discussion that in neighborhoods of the eigenvalues of H, the kernel K(A) should be replaced by a kernel K’(A) with shifted poles, in the manner specified in Lemmas 3.14 and 3.15. Since the free plane wave ?P,(x, k) has only a scattering channel component, 9(x, k, A) does not depend on whether we use K(A) or K’(h), and so we do not have to worry about avoiding the eigenvalues of H, . Q.E.D. 8. THE

EIGEN~TNCTION

EXPANSION

The distorted plane waves introduced in the previous section can be used as the eigenfunctions for expanding an arbitrary (column vector) function. Our proof of this expansion will be patterned after Ikebe’s proof [8] of the eigenfunction expansion for ordinary potential scattering with L, potentials. THEOREM

8.1.

Let V(x) be finite range and let f(x)

be an arbitrary function in

(L2(E))2. Then

(1)

the generalized Fourier transform t(k) = 1.i.m. E Y+(x, k)!(x) I

dx

(8.la)

off exists and{(k) E L,(M); the notation l.i;m. s *** a!x means limit in the mean of thefinction JlzlCR *. . dx as R -+ co. Also the generalized Fourier coeji?cients

(8.lb) off

exists and cf,

I fn I2 < 03.

POTENTIAL

(2)

The function /(x)

SCATTERING

BETWEEN

47

CHANNELS

can be expanded as

+(.x) = 1.i.m. JM Y(x, k){(k)

dk + l.i.;i.. 5 Y,(x)~~, n=l

(8.2)

where 1.i.m. Jhl -1. dk = l.i.m.R.+m,E+ JC
n=l

(8.3a)

l/a I2

and, for y(x) E (L2(E))2, the generalized Parseval equality: (9, f> = s, glW/(W (3)

dk + 2 $$n v.

(8.3b)

.

The full Hamiltonian &’ has the representation

X~(X) = 1.i.m.

sM

1k j2 Y/(x, k)+(k) dk + 1.i.i.. 5 E,Y&)$

.

a=1

(8.4)

Proof. Let E&) be the resolution of the identity associated with &’ and let PB be the projection onto the subspace corresponding to the discrete spectrum of X. Then 9 = (1 - P) is the projection onto the subspace of continuity of 2, and we will show that PC = Pat, i.e., that A? has no singular continuous spectrum. We assume for the moment that / (x) E (C,,m(E))2 and first consider the expansion of gcf in eigenfunctions of &‘. We start from the fundamental relation [17]

= (--1/2ni) I!$ s” (+, (W((p + ic)‘12) - a(& - ie)l/z)){) a where we will equation [ 161

always choose ,fI > 01 > 0. Employing (A2 - X2) W(A) W(X) = -[.qA)

the first

- a(x)],

dp,

(8.5)

resolvent (8.6)

we can rewrite Eq. (85) as:

Ntp, (E(P)+ W - 0))f> - (+g(EC4+ JY~- Wtp)l = (l/x) lii 1” l ({, W(& + ic)‘l”) a(& ‘2

- i+/3{)

dp

(8.7)

48

DASHEN,

Writing a&

HEALY

AND

MUZINICH

- ie)l12 in terms of the resolvent kernel ‘3, Pq(p - i~)“3)w>

= Gq(p + i+‘“)*f)(x)

(8.8)

= f @:*(x7Y, (P + w’2>jf(Y>

dY,

we can express the inner product in Eq. (8.7) as II W(cP - W2)ppl12 =

dzf +(x) @(z, x, (p + k)“3

f/++s,

@*(z,

Y,

(CL+ z-c)““) f(Y), (8.9)

where the orders of integration have been freely interchanged, since the absolute convergence of the integral is assured by the fact that @ satisfies the strong Carleman condition, Eq. (6.8). Let us write out @ in terms of its columns: g=(z, x, A) @*(z, y, 4 = @(x, z, 4 @+(y, z, a = @“‘(x, 2, A) P’+( y, 2, A) + sP(x, z, A) c!P+( y, z, A).

(8.10)

With this expression we can rewrite Eq. (8.9) as: II WP

- w2)pII z.r s, dx I, dy / +(4 is, L@‘)( x, z, (p + i-5)““) CP’+( y, z, (p + i~)l/~) + 2+)(x, z, & + i~)‘/~) +)+( y, z, Cp + ie)‘l”)] dzl f’(y).

(8.11)

We will show that the second term of Eq. (8.11) does not contribute to the righthand side of Eq. (8.7). Using the integral equation (6.11b) for @‘tc)we have: IS

@‘(x,

z, A) @‘(“+(y, z, A) dz

= / s, 1[@‘(x.

I

z, A) + s, %x, x’, 4 VW)

&‘(x’,

z, 8 dxf]

. [B:)+( y, z, A) + fE S$!+(xc, z, A) V-(x”) @+(y, x”, A) dx”/ dz ( G SUP YEE

II Go@,*>411”[I + s, I %, x’, A)~W>l dx’

+ fE I Wx”)

@+(Y,

(8.12)

x”, 41 dx”

+ s, I @(x,x’, A) Wd>l dx’ SEI ~(~‘7

@+(Y,

x”, 41 dx’]

< C - [I + j- [ @(x, x’, A) V-(x’)1 dx’] - [I + f, I V(x”) @+(y, x”, A)! dx”], E

POTENTIAL

SCATTERING

BETWEEN

49

CHANNELS

where we have used the strong Carleman condition on G, ; and combining Lemmas 3.18 and 6.2 with the assumption that V E L,(E) and V(x) = O([ x [-5/2--r) for j x 1 > R, R large, we can show that the integrals in the last Iine of (8.12) are bounded jB I @(A-,x’, A) Wd)l

dx’ < [i,,,,,

/ @(x, x’, h)j2 dx’]l”

+ S!r!,>R

. 11Y 11

/ qx, x’, A)] (dx’/l x’ /5/‘z+q

(8.13)

0 and A2 # En . Thus the integration in Eq. (8.12) is absolutely convergent and by Fubini’s theorem this justifies the interchange of the orders of integration in going from the second to the third line in Eq. (8.12). For X2 = r-1+ k the bound, Eq. (8.13) holds uniformly in OL< p < ,f3, and substituting Eq. (8.12) into Eq. (8.11) we see that the contribution of the second term of Eq. (8.11) to the right-hand side of Eq. (8.7) is

11:~LBE[s,dxjE@f+(x) (jE~%, z,cp+ iey2) G 'jg jwBE [jE lpP(~)ldy + jE (jE I V.U x @+(y, x”, t) + i~)~‘~)ld.u” I f(y)1 &I2 dp

for f(x) E (Com(E))2. Remembering that the Fourier transform of 9%)(.x, z, A) is @(x, k, A) and using Parseval’s equality we can write the first term on the right-hand side of Eq. (8.11) as j, dx jE &J+(x)

[j, @)(x, z, (,u + iV)

= j dx j dy f +(x> [j E

=.c

E

M

@S)+(y, z, (p + ie)“3 dz] f(y)

&.x, k, (II + W2) Y+(Y, k, (P + W2)

dk 1jEy+k MC/ k I2 - d2 + c2

k, (P + W’“>f

(4 dx I23

dk] P(Y) (8.15)

50

DASHEN,

HEALY

AND

MUZINICH

where the interchange of orders of integration is justified by absolute convergence:

G s, 1+(x)1 dx j-, I +‘(Y)I dy [s, I Ax, k, U2 dk s, I gldx12 XEE

,< c, since @ satisfies the strong Holder define

condition

and f(x)

E (C0m(Q)2. Now we (8.16)

f%, 4 = sEy+(X>k 8 f’(x) dx.

By Lemma 7.2 and the fact that { (x) E (C0”(E))2 we know that /(k, A) is bounded and uniformly continuous in k E M and h = @ + ie)l12 for E > 0 and 0 < (Y< p < /3 < cc provided no bound states E, of X are included in [cu,Is]. In particular, from Eq. (7.20) we have:

f-k

(8.17)

I k I) = f&Q.

Combining Eqs. (8.11), (8.14), (8.15), and (8.16) with Eq. (8.7) we obtain: H<,4 @@I + E@ - Wtp) - l

=(lb-1 ‘j$j-” (Ik12 $2 + ct [s,

E2If&,

01+ W2)12dk] 4.

(8.18)

We can interchange the orders of integration and take the limit E J 0 inside the k-integration, with the same justification as used by Ikebe [8] (ouri(k, (I-L+ in)‘/“) satisfies the same boundedness and continuity properties as does his function @(k, (p + ic)““)):

(l/n)I!,?: s”a[s,,,k,2 $2 + E2 If%Gc+ +““>I”dk]4 - ufM [Ilt$(+-)lo(Ik12442 + 9 Il&(k, (p + k)‘~“)l” dp] Finally using the continuity ofi(k,

dk.

(8.19)

& + ic)l12) and Eq. (8.17) we obtain:

H(fi (E(P) + -RB - 0)) f> - l

-1 ~l/e<,k,~l/z MkN2 dk2 provided no bound states of S are included in [a, /I].

(8.20)

POTENTIAL

SCATTERING

BETWEEN

CHANNELS

51

Taking iy.- /3, Eq. (8.20) reduces to

crp, (E(P) - m - WP>= 09

(8.21)

for any p (x) E (Com(E))2 ( w h ich is dense in (L,(E))z). Since [OI,81 is an arbitrary closed interval in 0 < p < co which does not include any of the discrete set of bound states E,, of &‘, we see that

is absolutely continuous in p > 0, except at the discrete set of points E, . Thus 2 has no singular continuous spectrum [6], and R =

Rat

+

Abound.

(8.23)

Taking the projection 9, onto the subspace of absolute continuity of z?, and using the unitary equivalence of the absolutely continuous parts of Z and & (Theorem 4.2) we can let 01-+ 0 and p 4 00 in Eq. (8.20) and obtain: 8.24)

Up to this point we have assumed that p(x) E (Co”)(E))2. The extension to f(x) E (L2(E))2 is standard [8, 181, i.e., we can show that p(k) exists and lies in L,(M) if we take the limit in the mean as specified in Theorem 8.1 in forming p(k). Thus Eqs. (8.20) and (8.24) remain valid when/(x) E (L2(E))2. The remainder of the proof is identical to Ikebe’s proof [8] of the corresponding statements for ordinary potential scattering with L2 potentials. Of course in our case we must include the positive energy bound states of .x? in sums over eigenvectors. Q.E.D. In Section 4 we proved the form of asymptotic completeness which Simon [6] calls “Kato asymptotic completeness.” We now have the stronger form: THEOREM 8.2. For Vfinite range the wave operators W, satisfy Simon’s “asymptotic completeness,” [6] i.e., &in &ac

=

Aout =

=

(hbound)l.

Aac

,

(8.25a) (8.25b)

Proof. We have established Eq. (8.25a) in Theorem 4.2, and for Y finite range we have Eq. (823), which completes the proof. Q.E.D.

52

DASHEN, HEALY AND MUZINICH

We will next show that the mapping from (L,(E))2 to L,(M) defined by Eq. (S.la) is onto L,(M); our argument is patterned after Simon’s proof [6] of the corresponding statements for ordinary potential scattering. We will then use this property to establish the complete characterization of the domain 9(Z) of selfadjointness of the Hamiltonian X in terms of the eigenfunction transform, Eq. (8.1). For arbitrary p (x) E (L,(E))2 we again introduce the Fourier transform f&k) of its scattering channel component by Eq. (7.17). Since the wave operator Wmaps the absolutely continuous subspace of YU onto that of X, we expect that for any free wave packet f (x), the wave operator W- should transform the part of it which behaves like the free plane wave Yf(x, k) into the distorted plane wave Y@, WTHEOREM

8.3.

Let V befinite range and+E (L,(E))2. Then

and

6-$,(k)

= f%),

(8.26a)

($)(k)

= h(k).

(8.26b)

Proof. If p $ RaC, then both sides of Eq. (8.26a) vanish, since Abound = (Aac)‘and pat W- = W- . Thus we need only establish Eq. (8.26a) for/~ ABC. Let S = (+E& (p(k) h as compact support in some set (k ( cy < I k I2 < /3)). First prove

Ctpt w-9) = j-Mf%l

(8.27)

5,(k) dk

for { E S and 9 E (C,,m(E))2. By Theorem 4.4 (f, W-9) = (f, Pay) + i h$ Jo-m (f, ei‘@?e-ipUtB,g)

eCtdt;

(8.28)

and using the generalized Parseval equality (8.3b) for the matrix element in the t-integral,
imtye-ixutg

= JMit(k)

sg) = J,f”l(k)

ei”‘t[&xy](k)

dk

eikzt [JE !P(x, k) T(x) e-iJ$utPsy(x) dx] dk,

(8.29)

we can write the t-integral as --li (f, eistV”e-iput9s.g)

est dt

s 0

= J:w dt JM dk s, dx/t(k)

Yf(x, k) ,(x)(e-itf~u~k2+iE’~~~)(x).

(8.30)

POTENTIAL

SCATTERING

BETWEEN

CHANNELS

53

The integrand is bounded by 1fs(k)l - 1Y+(x, Ic) V(x)1 - II &,(k)llI e-et and is absolutely integrable in x, t, and k since f(k) E L,(M) vanishes for 1k I2 < a and for 1 k I2 > /3 and /I V(x) Y(x, k)l[, IS * b ounded in 01 < [ k I2 < /?. This allows us to interchange orders of integration in Eq. (8.30), and since 9 E (C’m(E))2 we have :

=

s

M dk E dxf +(k) y/+(x, k) V(x)(-i)[Zu s

- k2 + ic)-’ .9&x)

dyf +(k) Y+(x, k) V(x) Sfu*(x, y, (k2 + ie)lj2) 8&y).

(8.31) The integrand is bounded, independent of E, by

l/WI

* I ~(4 ‘I/(x, @I I %gWlll x - Y I

which is absolutely integrable in x, y, and k. So we can interchange orders of integration again, and take the limit E $0 inside the integral; and Eq. (8.28) becomes:

= I ,j

(8.32)

+(k) @o(k) 4

where we have used the integral equation (7.3) for the distorted plane wave Y(x, k) and Eqs. (8.la) and (8.3b). Of course in neighborhoods of eigenvalues pL, of H, we must use a modified integral equation for Y(x, k), but the only change will be in the confined channel piece of J gUVY, which is irrelevant because of the presence of the projection P’s . This establishes Eq. (8.27) for 9 E (Com(E))2, and since Corn is dense in L, the equation (@f

lo(k)

= f%V

(8.33)

holds for tp E S. Furthermore, the set S is dense in &c, and using the continuity of the Fourier transform, Eq. (7.17), the generalized Fourier transform, Eq. (&la), and the wave operator W-+, it follows that Eq. (8.33) holds for any p E Rae,which com-

54

DASHEN, HEALY AND MUZINICH

pletes the proof of Eq. (8.26a). Then using the partial isometry of W- and Eq. (8.26a) we obtain (8.34) which is just Eq. (8.26b). COROLLARY

8.1.

Q.E.D.

The generalized Fourier transform, Eq. (8.la), is onto L,(M).

Proof. The proof follows Simon [6], with only modification due to the fact that our wave operators are partially isometric: Since W- is partially isometric, W-tw-

= 8,,

(8.35)

the operator Wmt is partially surjective: {W-+f 1, I f E tL(E))21 = L(E).

(8.36)

And since the Fourier transform, Eq. (7.17), is surjective we have {@$%I

I f E GtE)Y~

(8.37)

= L&W,

or, using Eq. (826a), c+ I f E tL2bwl

(8.38) Q.E.D.

= L2W).

It follows from this corollary that we can write “strong” formulas for operators in the eigenfunction transform space p(k) E L,(M): THEOREM

8.4.

Let V be$nite range and+ E 9(X).

Then

= I k 12ftW,

(8.39a)

(w)(k)

= eii~l*~(k),

(8.39b)

(%$%W

= %Q>jtk>,

(8.39c)

~~0 and

where 9(Q) = 1 on {k 1 1k I2 E J2} and Y(Q) = 0 on the complement. For the proof see Simon [6]. It is now easy to establish the complete characterization self-adjointness of H. THEOREM 8.5.

ifk”/(k)

Let V be finite range. Then E L,(M) andx,, E,,2 I$m I2 < co.

f

E B(S)

of the domain g(Z)

of

= .9(.?Q if and only

POTENTIAL

SCATTERING

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CHANNELS

55

Proof. By Corollary 8.1 the generalized Fourier transform, Eq. (8.la), is surjective, and the remainder of the proof is identical with Ikebe’s proof [8] of the corresponding statement for ordinary potential scattering. Q.E.D.

Finally we will establish the analog of the usual connection between the distorted plane waves and the scattering amplitude. We define the T-matrix by: T(k, k’) = s, Yf+(x, k) V’-(x) Y(x, k’) dx (8.40) = (eg$))(k’)*

for all k, k’ E M. LEMMA 8.1. If V is finite range, then T(k, k’) is well dejined and uniformly continuous in k E M and any compact domain of k’ E M.

Proof. By Theorem 8.4 Y(x, k) is bounded and uniformly continuous in any compact domain of k’ E M. The fact that V is finite range is then more than sufficient to ensure that Yf+(x, k) V’(x) Y(x, k’) E (L#))a and is uniformly ,!&ontinuous in k E M and any compact domain in k’ E M. Q.E.D.

For k2 = k’2, T(k, k’) is the usual on-shell T-matrix in the sense that, formally, it is related to the S-matrix by S = 1 - 2rriT6(Ef - Ei). We make this relation mathematically precise in exactly the same way as Simon [6]. THEOREM8.8. Assume V is$nite range. Let f(x), g(x) E S (where S is Schwarz space) and let their Fourier transforms f,(k) and j,(k) have compact supports in { k 1CY.< 1k I2 < /3>with no bound states k2 = E, included in [CX,p]. Then (f, (S - 11, g) = --hi

JM dk lM dk’fO(k) &,(k’) T(k, k’) 6(k2 - k’2).

(8.41)

The proof is given by Simon [6]. In our case the restriction that no eigenvalue E, of &’ be included in 01 < j k I2 < p is actually unnecessary because the distorted plane wave Y(x, k) which appears in the generalized Fourier transform is uniquely specified for all k E M, and T(k, k’) is continuous at the eigenvalues kt2 = E, . Furthermore the addition of a bound-state eigenvector Yn(x) to the distorted plane wave Y(x, k) at k2 = E, does not change the on-shell T-matrix: sE Y,(x, k) r(x)

Y,(x) dx = 0,

k2 = E,,

by Lemma 4.3, Eq. (5.12b). However as Simon [8] has pointed out, the result, Eq. (8.41), is already sufficient from an experimental point of view.

56

DASHEN,

HEALY

9.

AND

MUZINICH

DISCUSSION

As we mentioned in Section 1, there is no difficulty in including in the potential V(X) a diagonal scattering channel term Y&V). All of our results carry over unchanged if we impose on U,(X) the same conditions as were required of V(X), replacing the asymptotic condition V(X) = O(l x j-5/2-E) by U,(X) = O(j x 1-3-c) for large x The changes required in the proofs are slight and can be constructed easily from Ikebe’s proof of the eigenfunction expansion for ordinary potential scattering. The generalization of our results to any finite number of scattering channels is also straightforward. There are several places where our results might be able to be improved. For example it might be possible to eliminate the smoothness conditions V(X) = O(l x 1-5/2-E) and Holder continuous by employing a different approach, e.g., Simon’s method [6] of treating Hamiltonians as quadratic forms. On the other hand our proof of the eigenfunction expansion relies heavily on the assumption that V is Unite range, and we see no way to weaken this requirement. Our conditions on the cotined channel Hamiltonian could be greatly improved simply by reformulating them in a way which makes more transparent the class of confining potentials U,(x) for which they are satisfied. It would be desirable to know for what values of 01 > 0 the potential U,(x) = 1x 1. satisfies our requirements, and in particular if the linear potential U,(x) = I x ] is included in our class. Nothing in our analysis has ruled out the possibility of bound states degenerate with the continuum, and in general we know that they can be present. In fact for the particular example of an ii&rite radial square well confining potential and a finite radial square well off-diagonal potential we have explicitly constructed [9] positive energy bound states. They are orthogonal to the continuum states and are no more pathological than the confined channel eigenstates of the unperturbed Hamiltonian tiU . On the other hand, while the positive energy bound states of XU have an accumulation point at E = + co, we do not expect the positive energy bound states of the full Hamiltonian X to have such an accumulation point. That is, we expect that the Hamiltonian X will have only a finite number of normalizable eigenvectors. We have not proved this in general, but we would certainly be surprised if it were not true for the three-dimensional harmonic oscillator and shocked if it were not true for the infinite square well. For the infinite radial square well example mentioned earlier we have established [9] that the number of bound states for each partial wave is finite. In the case when the number of bound states is finite, the density of scattering states Y(x, k) is much greater than the density of free plane wave states !Pf(x, k). Consider the space S of function p(x) E (JL,(E))~ for which f3(x) = 0; then a nontrivial subspace of S is spanned by the distorted plane waves Y(x, k), but S is

POTENTIAL

SCATTERING

BETWEEN

57

CHANNELS

the orthogonal complement of the space spanned by the free plane waves Y,(x, k). For spherically symmetric potentials these statements are made more precise by our generalized Levinson’s theorem, Eqs. (1.19), which tells us that the total numbers of interacting states (scattering states plus bound states) equals the total numbers of states of tiU (free plane waves plus bound states of HC). We have established that the Hamiltonian 2 defines a perfectly sensible quantum-mechanical scattering system. We can now ask which other properties of ordinary potential scattering are also satisfied by these new systems. For example, does the on shell scattering amplitude, Eq. (8.41) satisfy a dispersion relation? Are there general relations, analogous to Bargmann’s bound for ordinary potential scattering, between the potentials (diagonal and off-diagonal) and the number of bound states ? To what extent does the scattering data determine the potentials ? These are all totally open questions, and work on them is in progress. In addition it would be satisfying if the generalized Levinson’s theorem, Eq. (1.19), could be derived directly from the existence and completeness of the generalized wave operators, as has been suggested in other contexts by Jauch and Polkinghorne [19]. We can also ask to what extent our models can be made more realistic. The first step to take in this direction would be to treat the Schrodinger equation (1.2) (which fully incorporates the quark structure of the hadrons) directly without assuming that only a finite number of coefficients V,, in the expansion of V(r, , r, , R; r) are nonvanishing. This is not as difficult as it might seem. If V is square integrable, s dr, drs dR dr 1 V(r, , r, , R; r)l” = 1)V iI2 < 00 (the local potentials, Eqs. (1.6) are not included in this class), then the off-diagonal potential Y” in Eqs. (1.2) is relatively bounded with respect to =%l = (

K(rJ

+ K(rd 0

+ H,(R)

0 f&(r) 1

with relative bound zero:

II“J-II2 < II VII2* II Fyl12 by two applications of the Schwarz inequality. Thus if XU is self-adjoint, then so is X = XU + V. Furthermore, if G,(r, , r, , R; r,‘, r,‘, R’; h) is the kernel of the resolvent R, = (A2 - H&J - Hc(rB) - H,(R))-l and V(r, , r, , R; r) = V(Ol(rm , r, , R; r) V(l)(r), then the solutions to the coupled integral equations

+ s dr,’ du,’ dR’ dr’ G,(r, x W,‘, Ye(r)

, r, , R; r,‘, r,‘, R’; A)

r,‘, R’, r’) YJr’)

= 1 dr’ dr,’ dr,’ dR’ G,(r, r’X) V(r,‘,

r,‘, R’; r’) Ys(ra’,

r,‘, R’)

58

DASHEN,

HEAL.Y

AND

MUZINICH

associated with the Schrodinger equation (1.2) can be obtained from the solution of the simpler coupled integral equations

!qr) = ‘y,““(r) + 1 dr’ G,(r, r’, A) F(r’) ul,(r) = / dr’ G,(r, r’, A) P(r’)

Ys(r’), Y,(r’),

where G,(r, r’) = / dr, dr, dR dr,’ dr,’ dR’ V(O)(r, , rB , R, r) x G,(r, , r, , R; r,‘, r,‘, R’, A) . V(*)(ru’, r.‘, R’, r-0, F*(r) = s dr, dr, dR V(0)(ra , r, , R; r) Ys(r, , r, , R), F?(r)

= s dr, dr, dR V(OJ(r,, rB, R; r) ?PJr,) Ym(rB>ei(E-E--BJ1’*R.

For V(O) such that G8(r, r’, X) is a Carleman type operator and V(l) E 4 n L, and finite range, say, the structure of these integral equations is very similar to that of the integral equation (1.15); and one can reasonably hope to extract some of the spectral properties of the Hamiltonian X by methods based on those we have used to study Eqs. (1.11) and (1.15). Of course one eventually would like to study a relativistic model with inelastic as well as elastic channels. An example is the model of quarks confined at the end of a relativistic string which, in two space-time dimensions and with a particular interaction between strings, seems to be equivalent [20] to ‘t Hooft’s two spacetime dimensional color gauge field theory [3] of quark confinement. This is a much more difficult problem, but any general rigorous results which could be established for such a system would be of great interest. Finally we want to emphasize the perhaps obvious point that in any theory with collfining forces one must never try to treat the conmung potential as a perturbation. Perturbation theory can only be meaningful if the perturbation is in some sense small relative to the unperturbed Hamiltonian. Any confining potential will drastically alter free particle states and cannot be small relative to the free quark Hamiltonian. In our Schrodinger equations (1.11) if we had chosen sU to be the kinetic energy operator in both confined and scattering channels and tried to treat the confining potential U, as a perturbation, then the wave operators

POTENTIAL

SCATTERING

BETWEEN

CHANNELS

59

would not even exist. The complete Hamiltonian # is not unitarily equivalent to the Hamiltonian which is the kinetic energy operator in both channels. In field theory it would also be nonsense to try to treat the confining potential as a small perturbation; for one of the foundations of quantum field theoretic perturbation theory is the weak asymptotic condition which states that insofar as their matrix elements are concerned the interacting fields approach free fields in the asymptotic past and future, and in a field theory of confined quarks the quark field operators certainly do not behave like free field operators for asymptotic times, Thus in any field theory in which the fundamental fields correspond to confined quarks, one must impose the weak asymptotic condition only on those composite fields which represent the physical hadrons, and treat the interactions between hadrons (as opposed to the interactions between quarks) as a perturbation of the Hamiltonian for free hadrons.

APPENDIX

Here we will study the properties of the Green’s function for the three-dimensional harmonic oscillator; and we will show that the conditions we have imposed on the confined channel Green’s function in Section 1 are satisfied by this example. The eigenvectors of the three-dimensional harmonic oscillator Hamiltonian: Hc = (-P/2)

+ $[c+~x~~+ wz2x22+ w&c~]

64.1)

are products of simple harmonic oscillator eigenvectors: Lzw

=

@,l(x,) =

R&4 (n!2y’2w:‘4

@T&2)

(A.2a)

@T&2)9

exp( - w&/2)

H,Jw:‘~x~),

(A.2b)

with eigenvalues (A.2c) where H, are the Hermite polynomials. The spectrum of He is bounded below by +(wl + w2 + w3) > 0, each eigenvalue is of finite multiplicity, and the set of eigenvectors (A.2) is well known to form a complete orthonormal set in L,(B). The functions cD~,(x~)are bounded, independent of ni and xi, by [21]: CD,“(x) < 1.0!+1/2.

(A.3)

60

DASHEN,

HEALY

AND

MUZINICH

And for large n, Schwid has established the asymptotic forms [22]: (i)

0 d x < (2n + 1)1/2,

@n2(x) =[

+

,3/2@

x ]cos [y

.+

~)~n+~~(;~:‘(x2,(~H

])))I/2

1

- x(2n + 1)1/q + O(n-‘1q1’2 (A.4a)

(ii)

x > (2n + 1)lj2

@‘,“(x) M [ x

exp[n - x2(1 - (2n + 1)/x2)1/21 n! (x2/(2n + l))‘/” & 3/2n(n+1)(x2/(2n + I))‘/“(1 - (2n + 1)/x2)’ e [l

-

(1

-

1

qL)1’2]2n+1

M c. exp[-x2(1

(iii)

- (2n + l)/x2)112](x2/(2n + l))“[l + (1 - (2n + 1)/x2)““] . r~l/~(l - (2n + 1)/x2)‘/” (A.4b) 12n + 1 - x2 1 = o(nY3).

@,2(x) = [n3,2nn;;;;4,3 31,3]1’2;r cf) - 6”322’3)5+ q-p 6” + -*- + O(n-l)12

where 5=Fxf$

(A-1.

1

These estimates are sufficient to establish the strong Carleman GG(x,y, X) and the continuity of SEG,(x, y, A)>f(y) & forfE L,(E).

(A.‘W property

of

THEOREM Al. For Jixed X2 # pn the resolvent R,(A) of the three-dimensional harmonic oscillator Hamiltonian, Eq. (A. l), is Carleman type and its Green’s function satisfies the strong Carleman condition

IIGc(x,-,4ll < c,

(A.5a)

POTENTIAL

SCATTERING

BETWEEN

61

CHANNELS

and the asymptotic conditions

for

II Gc(x, *y411 < O(l x r19>

Sup II Gc(x,.>A)11< O(l X 1(-1+c)/2)

1xl + co,

for* h2 + - co,

(ASb)

O
XXE

(A.%) Proof.

By Lemma 3.1 we need only prove that

c I &k412/lA2- Pn I2G c 1 &(X>/Pn)2< c

b4.6)

n

n

in order to establish the strong Carleman property; and an upper bound on the asymptotic behavior of the sum on the left-hand side of Eq. (A.6) will be an upper bound on the asymptotic behavior of II G,(x, ., h)l12. Let us f?rst consider the part of the sum for which ni > n,, for i = 1,2, 3 and n, large. Using the bound (A.71

and the factorizability of the wave function, Eq. (A.2a), we can bound the sum by a product of 3 sums, (A.81

and consider each of the sums in the product separately. It will be convenient to define the variable 2% + 1

f-=-----y-. Xi

64.9)

Then we can divide each sum into 6 pieces,

c @k(xJnf’3 =j c +(l+.,>T&,n?“, +~1+.,n?/‘,~~l-.,nl/3~ n$ao 7>1+E

f c 4,

I

E

I

(A.10)

j=l

and use the bound, Eq. (A.3), and the asymptotic forms, Eqs. (A.4a:b), to bound

62

DASHEN,

HFALY

AND

MUZINICH

each Zj . Only ZI and I, can contribute for xi2 < 2n, + 1; using the asymptotic form, Eq. (A.4a) we find:

for xi2< (ho + Q/(1+ 4, 115 c r>l+Ec n,“Yl- (llr>)-1’2 < I; . x_,,6 for xi2 > (2n, + l)/(l + l ); f2SC

c (l+E)>T>(l+E/n"'9

’

V6

(1

-

y2

1

q~~,,,~,,)

for

xi2 < (2n, + 1)/(1 + E),

for

xi2 > (2~2,+ I)/(1 + e),

$G

(1$,,,1,2@

lla)

I

1: - (x3-W

(A.llb)

The remaining terms can contribute only for xi2 >, 2n, + 1. To estimate I, we need only the bound, Eq. (A.3): Z3 5 C

C p,s I/Iz~‘~ w CrQ13 p-rl
c

. (x.2)-1/3

1

9

xi2 3 24 + 1.

(A.1 lc)

To estimate Z4, I, , and I6 we employ the asymptotic form (A.4b) and obtain: l+'@

4-

exp[-xs2(1

1

-

r)l/2]

exp(-x,2(1 l-E/n;'3

1

[

= 201/2r=l--E c

(1 _

r)1/2]2ni+l

r)l/2

- r)li2 - ((xi2r - 1)/2) ln r + xi$r ln[l + (1 - rY21} ((x+ - 1)/2)7/s(l - r)l12 (A.lld)

lmcff” exp{ -xt(l - r)3/2(l - (l/3))} (~,~/2)‘91 - r)lj2 Q=l--E

Pac.

< C(x&lIs

S:_, (1 frjl,2

< c * (X$2)7l6

c

exp[-x,2(l

5 f
+

2(21/2) 77n:/6(1 -

T=l--E

z N

r-ni[l

sE

x.2

xi2),2n,+

1;

- r)““] r-“[1 + (1 - r)1’2]2ni+1 2(2’/3 7rn:f6(1 - r)l12

z r-V (x,2r)71s

5 C - exp[--E1/2x,2]

exp [ - 3 xi2(1 - r)3/2]

+ (1 - rY’212ni+1 expI-x.2c1 t (1 - r)l12 Xi2>2n,+l;

_ rj1,21 dr (A.lle)

1

POTENTIAL

SCATTERING

BETWEEN

CHANNELS

exp[-xi2(1

- r)1’2] reni[l + (1 - r)1’2]2nS+1 2(2112)z-n,l/“(l - r)lj2

63

and I6 es

c

(2n&)lsi2

- 2~) xi”]

< Cexp[-(1

- 2~) xi21 s’

(A.1 lf)

C rpni O
(2no+l)lz,2

5 C * (xi”)“” exp[-(1

- 2~) xi”]

xi2 > 2n, + I.

Putting Eqs. (A.1 1) together with Eqs. (A.lO) and (A.8) we fmd (A.12a) and, since for / x2 j -+ cc we can always orient the axes so that all three ] xi2 / -+ co,

Then using the fact that @,((xJ is bounded independent of ni and xi and each @,i(xJ decreases exponentially as xi2 -+ co we see that the remaining contributions to the sum in Eq. (A.6) are also bounded and are dominated by the behavior Eq. (A.12b) for ] x2 ] -+ co. This completes the proof of Eqs. (A.5a) and (A.5b). To establish the bound Eq. (A.5c) we note that for X2 < 0 we can replace the uppser bound Eq. (A.6) by (A.13) and, by essentially the same argument used to establish Eq. (A.5a), the sum on the right-hand side of Eq. (A.13) is bounded for 0 < 6 < 4. Q.E.D. Theorem A.2. Let Gc(x, y, A) be the Green’s function for the three-dimensional harmonic oscillator Hamiltonian and let f (x) E L,(E). Then for fixed x2 # pn

I G&TY, 4f(u> E

is Hiilder continuous in x. 5951IO41-5

dy

(A-14)

64

DASHEN,

Proof.

HEALY

AND

MUZINICH

We begin by using the Schwarz inequality and Eq. (2.1Ob) to obtain

the bound:

We choose the orientation of our axes so that dx = x’ - x is in the x1 direction, and let n, be a very large positive integer. Then we can break up the sum in Eq. (A. 15) as follows:

where (A.16b) with Dj defined by 0 < n, e &J ,

j=

(2n, + 1)/x,2 < r < E,

j = 6;

E < r < (1 - E),

j = 5;

(1 - 6) < r < (1 - •/n,“‘~) (1 - +z~‘~) < r < (1 + •/nf’~),

j = 4;

(1 + +z:“)

j = 2;

< r < (1 + E),

j=

j=

(1 + 4 < r,

0;

(A.16c)

3;

1;

and I = (2n, + 1)/x,2.

(A.16d)

We observe that I, c * (Ax)2

(A-17)

since IO is a sum of a finite number of terms, each of which satisfies this inequality. To estimate the remaining contributions we use the large 12behavior of the eigenvectors given by Eqs. (A.4):

POTENTIAL

(i)

SCATTERING

BETWEEN

65

CHANNELS

For I > (1 - +I:/~) we have

P5nJXl) - @&l’N”- Inl(l -cl,r)]‘,2[co,(F - cos(F

x1@

+ w2j

- x,‘(2n + l)‘iaj]z,

(A.18)

where we have used

(

1-

!2 x1 ) xl2 j * (1 24 + 1 2n, + 1

for (LIx)~ < (2n, + 1); and we can bound Eq. (A.18) by:

with m > 1. Comparing with the computations dx small we have

in Eqs. (A.lla,

b) we see that for

Zl + I2 < c * (Llx)2/m

(A.20)

provided m >, 6 + E. (ii) For (1 - +z:‘~) < r < (1 + +z;‘~) and dx = o(~/(x,~)~/~) we have 5 = (x,2)213(1 - r) < O(E) and by Eq. (A.4c) we find -

P&l)

- @&,‘)I2 d $$g

Gw2 5 &

m2,

(A.21)

and consequently: 13

< c - @x)2.

(A.22)

On the other hand for E > dx 2 o(.s/(x~~)“/“) with x12 large, we obtain from Eqs. (A.11) the bound

c L(X) - i%x’)12< c * (x12)-l/6< c * (dx)li4. n I A2- PnI2 (iii) For P < (1 - 6/n213),x, is large and (2n + 1)/x12 w (2n, + Mh for dx small. Thus using Eq. (A.4b) we obtain

FRa,txl)- @&,‘)I” .s @n,(x,> X12U - rew2

(A.23) + w2 (A.24)

66

DASHEN, HEALY AND MUZINICH

and referring back to the estimates, Eqs. (A. 1 Id, e, f), we see that this is sufficient to prove (A.25) 14 + 15 + If3 < C(W. Putting together Eqs. (A.17), (A.lS), (A.22), (A.23), and (A.25), and noting that the right-hand side of Eq. (A. 15) is bounded by a constant, independent of x and x’, we find that JE G,(x, y, h) f(v) dy is Holder continuous with exponent 6 + E, provided h2 # p,, . Q.E.D. Finally we need to put a bound on the modulus of the harmonic oscillator Green’s function in terms of the free particle Green’s function. We will restrict ourselves to the isotropic oscillator, w = w1 = o2 = wg. It will be convenient to introduce the time-dependent Green’s functions K(x, y, t) as solutions to the Schrijdinger equation HW,

Y, t> = Wit>

m

(A.26a)

Y, t>

which satisfy the boundary condition K(x, y, 0) = 63(x - y). The usual time-independent Green’s function the Fourier transform of K(x, x’, t):

(A.26b)

G(x, y, (JY)‘/~) = (E - H)-l

is

(A.27)

G(x, y, E1/2) = JamK(x, y, t) eiEt dt.

For the particular cases of the free particle Hamiltonian Hf = -V2 and the threedimensional isotropic harmonic oscillator Hamiltonian H, = -V2 + w2x2 the time-dependent Green’s functions are known explicitly [23]: K,(x, y, t) = (2~it)-~/~ exp[-(x - ~)~/2it], (A.28a) J&(x, y, t) = (w/2vi sin CO~)~/~ exp{--w/2i sin wt[(x” + v2) cos wt - 2x * y]}. (A.28b) For the harmonic oscillator Green’s function the contour of inegration in Eq. (A.27) must be distorted into the lower half t-plane to avoid each singularity of (sin wt)-l at t = mr/w for m > 1. THEOREM A.3. For fixed A2 # p,, the three-dimensional oscillator Green’s function G,(x, y, A) satisfies the inequality

I G,(x, y, A)\ < , xeyy

, + cle-(w/4)(~--y)z <

where C, and C, depend on A2 but not on x or y.

isotropic harmonic ‘2

Ix-Yl'

(A.29)

POTENTIAL

SCATTERING

BETWEEN

67

CHANNELS

Proof. Temporarily restrict E = X2 to the upper half energy plane Im E > 0. For Re E < 8~ we can rotate the contour of integration 90” into the lower half t-plane; the integrand is exponentially damped as i Im f + co, so there is no contribution from the quarter circle at co and we can write G, as:

Gc(x, y, E1j2) = jo-im ( 2~i ln wt )“I exp 1 2i sin --w wt x [(x2 + y2) cos wt - 2x . y]l eiEt dt m =.i

0

w

2~ sinh wl

(

x [(x2 + y) cash wt - 2x . y] eEt dt.

(A.30)

I

Changing variables to T = sinh wt and using cash wt > I and (eReE.t/cosh it) for Re E < w we obtain 1 Gc(x, y, El’31 < sa (w/277~)~/* exp[( -w/27)(x

< 2

- Y)~](~/w) dr

0

= Ul4(lll

x - Y I>

for

Re E < w.

(A.31)

For Re E > w we use the fact that K,(x, x’, t) is periodic in t, K&G Y, t + (2n7+))

(A.32)

= e-3nniKc(x, y, t),

to write the resolvent kernel G, as Ge(x, y, E112) = j’”

Kc(x, y, t) eiEt dt

0 e2”““E/“‘-‘“,2”)“]

E/W =s

KG(-y,

y, t) ,$-Et +

1:“““”

[l

A

Kc(x,

y,

t) eiEt

&,

e2ni((Elw)-(3/2))]-1

0 (In+t)/w X

s EIW

K,(x, y, t) eiEt dt.

(A.33)

The poles of G, at E = pn are now exhibited explicitly. If we choose E to be pure negative imaginary and small, then we can distort the contour of integration in

68

DASHEN,

HEALY

AND

MUZINICH

the second integral to a straight line from e/w to (27r + c)/w. The second integral is then bounded by (2n-+l)/w Kc(x, y, t) eiEt dt IS -ilrllw I

= ~(2g

jo2r(

t’ cash

sin

I

)3’2

I E j - i cos t’ sinh 1 E 1

h/2

* exp (sin t’ cash j E ) - i cos t’ sinh ) E J) * Kx - Y)” + (x” + Y2) . (cos t’ cash 1E I + i sin t’ sinh / E 1 - l)]/ * ei(E@)(t’-iC)$- 1 < ($)3’z(&)3’2

cxp [

--w(cosh I E I (x2 + y') - 2x . y) 2(1 + sinh2 I E I)

1

27r sinh(r Im E/a) elrl(ReE/w) * TIT e(-nlmElw’ x Im E/w exp [ -w((x

523($f-)1’2($)3’2

- Y)” + (‘2/2)(x2 + Y”)) 20 + c”)

1e,r,ReE,os (A 34)

For the remaining integral in Eq. (A.33) we have:

--ijcl/w Kc(x, x’, t) e=tdtl II

= 1jo+‘E’(2ns?ht,)3’2

0

[(x2 + y2) cash wt - 2x * y]/ eEt’lw $

* expI& < ; jo-

)

(2-)3’2

. exp e t

[(x2 + y2)(1 + .2)1/2 - 2x . y]/ ‘2s;;;

dT

<

5

elr](ReR/w)

1

cash E 5

1 1x - y / *

(A.35)

Choosing j E j = (w/Re E) x 1 in Eqs. (A.34) and (A.39 we find: 1G,(x,

y,

El/‘)1

<

g

X

for Re E > CO.

, x

_

e (+rnexp

:

, +

(1

-

(+(x

@i((E/w)-(3/2)))-1

- y)2)

(-E)l”

(A.36)

POTENTIAL

SCATTERING BETWEEN CHANNELS

69

Combining Eqs. (A.31) and (A.36) we have

/Gc(x,~,Ql <-&-,,A,, +CIexp[T(x-y)2] < ,x2y,

(A.37)

for fixed X2 # CL,,in the upper half plane, where C, and C2 are independent of x and y. Finally, since G,(x, y, (E)1/2) is a real analytic function of E, the same bound is also valid in the lower half plane. Q.E.D.

ACKNOWLEDGMENT We are grateful to Professor T. Regge for several enlightening discussions about potential scattering. Two of us (JBH and IJM) wish to thank Professor C. Kaysen for his hospitality at the Institute for Advanced Study where this work was initiated. One of the authors (JBH) is indebted to Professors B. Simon and G. Zuckerman and to his colleagues in the Yale theoretical physics group for stimulating discussions, and to Professor L. Thomas for a valuable communication. Another of us (IJM) thanks Professor B. W. Lee for hospitality at Fermilab where part of the work was carried out.

REFERENCES 1. See, e.g., J. J. KOKKEDEE, “The Quark Model,” Benjamin, New York, 1969. 2. For a recent review of the quark confinement problem see R. F. DASHEN, Quark confinement, in “Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies,” Stanford, August 21-27, 1975. 3. Field theories in 2 space-time dimensions which exhibit confinement include the nonabelian color gauge theory (quantum chromodynamics) of G. ‘T HOOFT, AU. Phys. B75 (1974), 461; and the Schwinger model, see A. CASHER, J. Koowr, AND L. SUSSKIND, Phys. Rev. DlO (1974), 732. 4.

5. 6. 7. 8. 9.

10. 11. 12. 13.

For a thorough discussion of ordinary potential scattering see R. G. NEWTON, “Scattering Theory of Waves and Particles,” McGraw-Hill, New York, 1966. A more mathematically oriented treatment of ordinary potential scattering is given by V. DE ALFARO AND T. REGGE, “Potential Scattering,” North-Holland, Amsterdam, 1965. B. SIMON, “Quantum Mechanics for Hamiltonians Defined as Quadratic Forms,” Princeton Univ. Press, Princeton, N. J., 1971. T. KATo, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin/Heidelberg/ New York, 1966. T. IKEBE, Arch. Rut. Mech. Anal. 5 (1960), 1. R. F. DASHEN, J. B. HEALY, AND I. J. MUZINICH, Theory of multichannel potential scattering with permanently confined channels, Phys. Rev. D 15, in press. See, e.g., F. Rrasz AND B. SZ-NAGY, “Functional Analysis,” Ungar, New York, 1955. A. YA POVZNER, Mat. Sbornik 32 (1953), 109; see also [8]. W. HIJNZIKER, HeIv. Phys. Actu 39 (1966), 451. G. TIKTOPOLOUS, Phys. Rev. B133 (1964), 1231.

70

DASHEN, HEALY AND MUZINICH

14. T. KATO AND S. T. KURODA, Nuovo Cimento 14 (1959), 1102. 15. S. T. KURODA, Nuovo Cimento 12 (1959), 431; I. .Z. Math. Sot. Japan 11 (1959), 247. 16. E. HILLE AND R. S. PHILLIPS, “Functional Analysis and Semigroups,” Amer. Math. Sot. Colloq. Publ. 31, 1957. 17. M. H. STONE, “Linear Transformations in Hilbert Space and Their Applications to Analysis,” Colloq. Publ. A.M.S., New York, 1932. 18. E. C. TITCHMARSH, “Introduction to the Theory of Fourier Integrals,” Oxford, London, 1937. 19. J. M. JAUCH, Helv. Phys. Acta 30 (1957), 143; and J. POLKINGHORNE, Proc. Cambridge Phil.

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