ObI T H E

CONNECTION

COEFFICIENT V.

S

BETWEEN

AND THE Buslaev

and

THE

THIRD

VIRIAL

S MATRIX S.

P.

Merkur'ev

The t h i r d v i r i a l c o e f f i c i e n t in quantum s t a t i s t i c a l p h y s i c s is e x p r e s s e d in t e r m s of the t w o -

particle and three-particle S matrices. 1.

Introduction

1. It has been c o n j e c t u r e d that the n - t h group i n t e g r a l in quantum s t a t i s t i c a l p h y s i c s can be e x p r e s sed in t e r m s of the S m a t r i x of a s y s t e m of n p a r t i c l e s and the S m a t r i c e s of the s u b s y s t e m s . At the p r e s e n t t i m e this h y p o t h e s i s has been j u s t i f i e d only for n = 2. In the p r e s e n t p a p e r a s i m i l a r r e s u l t is obtained for n = 3. In p a s s i n g , we a l s o d i s c u s s the c a s e n = 2. The second and t h i r d group i n t e g r a l s f o r Boltzmann s t a t i s t i c s can be e x p r e s s e d in t e r m s of the e n e r g y o p e r a t o r s by the f o r m u l a s

b2 ~

sp (e -~h -- e+h0),

(1. t)

In t h e s e f o r m u l a s , h and H a r e the e n e r g y o p e r a t o r s of s y s t e m s of two and t h r e e p a i r w i s e i n t e r a c t i n g p a r t i c l e s ; h 0 and H 0 a r e the c o r r e s p o n d i n g kinetic e n e r g y o p e r a t o r s ; and I ~ (a = 2 3 , 31, 12) is the e n e r g y o p e r a t o r of a t h r e e - p a r t i c l e s y s t e m in which the i n t e r a c t i o n of only the c~-th p a i r is taken into account. All the o p e r a t o r s a c t on s p a c e s for which t h e m o t i o n of the c e n t e r s of m a s s has been s e p a r a t e d ~ In 1937 Bethe and Uhlenbeck, a s s u m i n g the p a i r i n t e r a c t i o n p o t e n t i a l s to be s p h e r i c a l l y obtained an e x p l i c i t e x p r e s s i o n for b 2 in t e r m s of the s c a t t e r i n g p h a s e s (see [1, 2]). The c a s e not s p h e r i c a l l y s y m m e t r i c p o t e n t i a l s was c o n s i d e r e d much l a t e r . In this connection, we m a y i n v e s t i g a t i o n s [3, 4], in which this p r o b l e m was s t u d i e d in connection with the s o - c a l l e d t r a c e follows f r o m [3], in p a r t i c u l a r , that --

symmetric, of genera1, mention the formulas~ It

n

ei~t

v\s"

(1.s)

0

w h e r e a i is the e n e r g y of the bound s t a t e s and s(E) is the t w o - p a r t i c l e S m a t r i x . f o r m u l a can a l s o be r e p r e s e n t e d in the f o r m

Note that the t r a e e in this

s p ( s . .(~)~]ds(E) .. \ ~_~ "d lndets(E) d E

(1.4)

2. Of the p a p e r s devoted to the t h i r d group i n t e g r a l , we m a y mention those of Smith [5] and B e r e z i n [6, 7]. In d i s c u s s i n g the r e s u l t s of t h e s e p a p e r s , we s h a l l r e s t r i c t o u r s e l v e s to the c a s e of p e r f e c t l y e l a s t i c s c a t t e r i n g . More p r e c i s e l y , we shall a s s u m e for the t i m e being that the t h r e e - p a r t i c l e e n e r g y o p e r a t o r H and the t w o - p a r t i c l e o p e r a t o r h do not p o s s e s s bound s t a t e s .

Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya No. 3, pp. 372-387, December, 1970. Original artiele submitted March 27, 1970.

Fizika, Vol. 5,

9 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~/est 17thStreet, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $I5,00.

1216

Comparison of formulas (1.1)-(1.3) suggests

b~

--

u I fe-~Sp ( S * ( E ) ~ ar~

~ 2-~

(1.5)

) ~dE'

0

where "(E)

and s(g) and Sa(E) are the matrices

.z~

,~

=

S'(E)

for the operators

dE

&'(E)

dE

'

H and H a.

In fact, this formula is given by Smith [5], who was, apparently, tween the third group integral and the scattering characteristics.

first to study the connection be-

It turns out, however, that the relation (1.5) is devoid of direct meaning. Whereas its left side is defined by formula (1.2) quite correctly (see [8]), the trace in the integrand contains infinite terms and one cannot establish whether these cancel. These infinitives arise because the kernel of the operator S(E)-I (I is the identity operator) in the three-particle ease contains strong singularities on the diagonal corresponding to the processes of single and twofold scattering. Note that in the two-particle case the analogous operator s(E)-I is defined by a smooth kernel. 3. One might hope that the indeterminate quantity (S*(E)dS(E)/dE) admits regularization in some manner and that formula (1.5) then acquires a well defined and correct meaning. However, even this assumption is erroneous. In fact, there exist model operators, close to H in a number of respects, for which the trace Sp(S*(E) - dS(E)/dE) is defined and even vanishes, whereas the trace of the expression on the right side of formula (1.2) is definitely nonzero. An example is the Hamiltonian of a system of two particles interacting with a fixed center. In this problem the variables are separable and the problem reduces to an investigation of two-particle operators of the type h. The operator arises as a degenerate case of the three-particle Hamiltonian, in which two particles do not interact with each other and themass of the third tends to infinity. "~ This circumstance prompted Berezin [6, 7] to conclude that the right side of (1.5) must contain additional terms which can be expressed in terms of In s(E). The regularization of Sp(S*(E)dS(E)/dE)c was not discussed in [6, 7]. 4. In the present paper, anumber result: for functions ~o(E) that decrease

of whose results were announced in [9], we establish the following sufficiently rapidly on the spectrum of H the trace of the operator

[+(rI) ]o = and, in particular,

+ (H) -

+ (Ho)

- ~ , (,+(I4~)

-

~ (no))

the group integral b 3 in the case of perfectly elastic scattering,

is equal to

where the operators A(E) and A(E) can be expressed explicitly in terms of the two-particle S matrices. These expressions are rather cumbersome and it may well be possible to rearrange them in a more compact form. We should mention, however, that the operator A(E) is, in a certain sense, the simplest operator expressed in terms of the pair S matrices whose subtraction permits regularization of the trace of (S*(E)dS(E)/dE) c" In the derivation of the above formula, the particle masses and the interaction potentials are not assumed to be identical. The result can be readily generalized to the case when the scattering is not perfectly elastic and H has a finite number of bound states as well as a continuous spectrum (see the end of Section 3). The approach tWe

developed

here is applicable to both Bose and Fermi

statistics.

are grateful to F. A. Berezin for acquainting us with this model.

1217

5. In this p a p e r we have s t r i v e d to combine a d e s c r i p t i o n of the connection between b 3 and the S m a t r i x and a g e n e r a l s y s t e m a t i c e x p o s i t i o n of the d e r i v a t i o n of such a s s e r t i o n s . T h i s is a new method (and, p r o b a b l y , the m o s t simple) even for two p a r t i c l e s . Unfortunately, l a c k of s p a c e p r e v e n t s us f r o m giving a l l p r o o f s in full, the m o r e so s i n c e m a n y of them a r e f a i r l y lengthy. H e r e i s an outline of the p a p e r . In Section 2 we c o l l e c t i n f o r m a t i o n on the o p e r a t o r H n e c e s s a r y for the subsequent exposition of the m a i n m a t e r i a l . It is m a i n l y taken f r o m [10]. In Section 3 we c o n s i d e r a r e l a t i o n s h i p of the f o r m N

i~l

em

whose left s i d e goes o v e r into b 3 for ~v(E) = (q-3/2A3)e-P E. F o r m u l a (1.6) e x p r e s s e s ~+(E) in t e r m s of the t h r e e - and t w o - p a r t i c l e S m a t r i c e s . The c a s e of p e r f e c t l y e l a s t i c s c a t t e r i n g i s d i s c u s s e d f a i r l y fully but the final r e s u l t s a r e a l s o given f o r the g e n e r a l case. At t h e s a m e t i m e , the w e l l - k n o w n analogous f o r m u l a f o r b 2 is d e r i v e d , The a u t h o r s a r e g r a t e f u l to L. D. F a d d e e v f o r his i n t e r e s t in the i n v e s t i g a t i o n and for n u m e r o u s fruitful d i s c u s s i o n s . We a r e a l s o g r a t e f u l to F. A. B e r e z i n f o r v a l u a b l e d i s c u s s i o n s . A f t e r the p r e s e n t p a p e r had been sent to p r e s s , we b e c a m e acquainted with two r e c e n t p a p e r s of Dashen, S. - k e n g Me, and B e r n s t e i n [11, 12]. T h e s e p a p e r s r e v e a l d i f f i c u l t i e s connected with the i n t e r p r e t a t i o n of S m i t h ' s f o r m u l a . The e x i s t e n c e of the a d d i t i o n a l t e r m s (3.14) was not known to the a u t h o r s . The e x p r e s s i o n for b 3 in t e r m s of the S m a t r i c e s is a b s e n t m~d the content of the p a p e r r e d u c e s to a d e r i v a t i o n of the p r e p a r a t o r y r e l a t i o n s h i p (3.4) and to a d i s c u s s i o n of the question of the e x i s t e n c e of ~=~(E) = l i m ~(E + ic). t~0

2.

Properties

of the

Basic

Operators

2. We shall consider a system of three nonrelativistic pairwise interacting particles. The interaction is defined by potentials that depend on the relative coordinates of the particles and decrease rapidly as the particles move away from each other. From the mathematical point of view this system has been studied in detail by Faddeev [I0] ; an appreciable amount of the preparatory material of the present section is taken from [I0] without further citation. Besides the three-particle system, we shall consider systems of two particles. We shall assume that the motion of the center of mass has always been separated and the momentum representation and Jacobi coordinates will be used to describe the systems. The energ'y operator of a two-particle system h (or ha, where ~ takes the values 12, 23, 31 if it is necessary to specify the chosen pair in the three-particle system) acts on the Hilbert space L2(R3). The arguments of the functions of this space will be denoted by k (or, in accordance with the above convention, k~). The operator h = h 0 + v acts in accordance with the formula (h]) (k) =

h~(k)f(k)+ ~ v(k-- k')](k')dk', ho(k)= k~/2m,

w h e r e m is the r e d u c e d m a s s of the p a i r of p a r t i c l e s and v(k) is the F o u r i e r t r a n s f o r m of the i n t e r a c t i o n potential. The e n e r g y o p e r a t o r of the t h r e e - p a r t i c l e s y s t e m H a c t s on the H i l b e r t s p a c e L2(R ~) of functions of the two m o m e n t a k and p conjugate to the p a i r of J a c o b i c o o r d i n a t e s . T h e r e e x i s t t h r e e s y s t e m s of J a e o b i c o o r d i n a t e s that a r e connected to one another. If it is n e c e s s a r y to s p e c i f y the a c t u a l s y s t e m , Lhe c o o r d i n a t e s will a l s o be i d e n t i f i e d by a s u b s c r i p t ~. Then 1% (a = 23, 21, 12) is the m o m e n t u m c o r r e s p o n d i n g to the r e l a t i v e c o o r d i n a t e of the ~ - t h p a i r and pa is the m o m e n t u m c o r r e s p o n d i n g to the c o o r d i n a t e of the t h i r d p a r t i c l e r e l a t i v e to the c e n t e r of m a s s of the ~ - t h p a i r . The s u b s c r i p t a of the m o m e n t u m p t a k e s the v a l u e s 1, 2, 3 i n s t e a d of 23, 31, 12, r e s p e c t i v e l y . The e n e r g y o p e r a t o r H has the f o r m

H=

He+V,

V ~--- ~-a V~,

w h e r e H 0 is the o p e r a t o r of m u l t i p l i c a t i o n by the function

1218

[ma (~ = 23, 31, 12), na (a = 1, 2, 3) a r e the c o r r e s p o n d i n g r e d u c e d m a s s e s ] , and (V+/) (k,p) = ~v~,(k~, -- k~') f ( kJ, p~) dk~' (va is the F o u r i e r t r a n s f o r m of the i n t e r a c t i o n potential of the a - t h pair). If the t h r e e - p a r t i c l e o p e r a t o r tt or any entities e x p r e s s e d in t e r m s of H c a r r y the s u b s c r i p t a , this m e a n s that we c o n s i d e r the c a s e V = "Ca. We s h a l l f r e q u e n t l y c o m b i n e a p a i r of J a c o b i m o m e n t a into a single v e c t o r P = {k, p} E R G. If the v a r i a b l e k a is a c c o m p a n i e d by the a r g u m e n t s k~ (p~, p ( ) , this denotes the function r/it

k,2 ~-- -- p /

- - p 3 m, + m2

m2

= p/+ --p~, mt + m2

in the c a s e ~ = 12 and the function obtained by c y c l i c p e r m u t a t i o n of the s u b s c r i p t s f o r the o t h e r value of ~. A p r i m e d a r g u m e n t c o r r e s p o n d s to a n i n d e p e n d e n t c o n f i g u r a t i o n of the s y s t e m . T h e connection between the J a c o b i m o m e n t a is s u c h that k~(p~, p~) ~ k~. In what follows, it will be helpful to u s e the a b b r e v i a t i o n s ~ ~ k~ (p~, p~') and k,a k~ (p~', p~). 2. As r e g a r d s the F o u r i e r t r a n s f o r m v(k) [and va(k)] we a s s u m e that: 1) v(k) = v ( - k ) ; 2) t h e r e exist the continuous d e r i v a t i v e s D • up to o r d e r I x I = 5 s a t i s f y i n g

ID~v(k) l ~ C ( l + ] k t )

-~-~

1 0>~-;

3. The o p e r a t o r s (e + vr0(0)) , w h e r e r0(z ) is the r e s o l v e n t of h0, a r e i n v e r t i b l e on a suitable B a n a c h s p a c e (see [10]); 4. T h e o p e r a t o r s h(ha) have only a finite n u m b e r of e i g e n v a l u e s e~(e~), e+< 0, i = 1, 2, . . . , n(na). 5. T h e o p e r a t o r II has only a finite n u m b e r of e i g e n v a l u e s E i and E+ . ~ em --~,rain e~ ~.

The assumptions we have made allow us to use all the results of Faddeev's paper [10] and also of the authors' paper [8] without reservation. 3. Let r(z) and R(z) be the resolvents of h and H. We shall operate with the T matrices t(z) = v - v r ( z ) v and T(z) = V-VR(z)V. These are integral operators, whose kernels will be denotedby the same symbols.

T h e k e r n e l of t(z) a d m i t s the r e p r e s e n t a t i o n n

t(k,U;z)=

Z i=l

---~-^ t ;z). (P+(k)~(k') ~-t(k,k Z --

(2.1)

~i

T h e function
The function t(k, k'; z) is h o l o m o r p h i c in z on the plane with the h a l f - a x i s z _> 0 r e m o v e d and is a bounded d i f f e r e n t i a b l e function of its a r g u m e n t s on the c i o s e d plane with the cut z _> 0. Bounds f o r t(k, k'; z) a r e given in [101 and bounds f o r the d e r i v a t i v e s in [8]. Let T~(z) = T ( z ) - - L

T~(z). r

T h e k e r n e l s of the o p e r a t o r s Ta(z) have the f o r m

T~(P, P'; z) : t,(k:, k ~," z--p~/2n~)5(p.~--p~').

(2.2)

T h e k e r n e l of Tc(Z ) a d m i t s the r e p r e s e n t a t i o n

1219

To(P,P';z)

~.

@,(P)'O,(P') ~ !'~(P, P ,'. zS, ' z - - E~

=,

(2.3)

where .,(p)

=

(He(P) - E0,v~(p),

and ~ i a r e t h e o r t h o n o r m a l i z e d e i g e n f u n c t i o n s of H. Off t h e c o n t i n u o u s s p e c t r u m the k e r n e l T c h a s p r o p e r t i e s s i m i l a r to t h o s e of t h e k e r n e l t ( s e e [10, 8]). On t h e c o n t i n u o u s s p e c t r u m T c ( P , P ' ; z) p o s s e s s e s s i n g u l a r i t i e s of two t y p e s , w h i c h a r e m a n i f e s t e d by t h e e x i s t e n c e of f a c t o r s t h a t a r e s i n g u l a r on c e r t a i n m a n i f o l d s . T h e s i n g u l a r i t i e s of t h e f i r s t t y p e a r i s e f o r t h e s a m e r e a s o n a s t h e s i n g u l a r i t i e s of t h e k e r n e l s of Tc~(z) ( s i n g l e and t w o f o l d s c a t t e r i n g ) . W e s h a l l r e f e r t o t h e s e a s t h r e e - p a r t i c l e s s i n g u l a r i t i e s . T h e y e x i s t e v e n in t h e c a s e of p e r f e c t l y e l a s t i c s c a t t e r i n g and a r e c o n t a i n e d in t h e f i r s t two i t e r a t i o n s of F a d d e e v ' s e q u a t i o n s [10]. A l l o w a n c e f o r t h e s e s i n g u l a r i t i e s d e t e r m i n e s the f o r m of t h e o p e r a t o r s A(E) and A{E), (1.6), w h i c h r e g u l a r i z e t h e e x p r e s s i o n f o r t h e t h i r d g r o u p i n t e g r a l . W e s h a l l c a l l s i n g u l a r i t i e s of t h e s e c o n d t y p e t w o - p a r t i c l e s i n g u l a r i t i e s . T h e y e x i s t if t h e r e a r e p a i r bound s t a t e s and o c c u r in the s a m e f o r m in a l l i t e r a t i o n s of F a d d e e v ' s e q u a t i o n s . A l l o w a n c e f o r t h e s e s i n g u l a r i t i e s d o e s not i n t r o d u c e a t h r e e - p a r t i c l e c h a r a c t e r i s t i c into t h e c a l c u l a t i o n of b 3. It i s i m p o r t a n t t h a t the t w o - and t h r e e - p a r t i c l e s i n g u l a r i t i e s do not i n t e r s e c t ( s e e [10]). In o r d e r to d e s c r i b e the s i n g u l a r i t i e s of the k e r n e l T c ( z ) w e i n t r o d u c e the o p e r a t o r s W~(z) : Note that Tc(z ) = E

M~.~(z) - - 5~T~(z), where M~(z) - - 8 ~ V ~ - V.~R(z)V~.

W"t3(z)" T h e o p e r a t o r s Was(z) can be e x p r e s s e d a s a sum: W~(z) =

T~,(z)Ro(z)T~(z) + ~ ' , ~ ( z ) .

--

T h e k e r n e l s of TaRoTt~ h a v e t h e f o r m

(T~RoT~) ( P , P ' ; z ) = t.

(2.4)

The singular factors of the kernels of TaRoT s which occur as terms in Tc(z ) are responsible for the threeparticle singularities of Tc(Z ). Three-particle singularities are also contained in the kernel of Was but they have only a logarithmic nature. This permits us to ignore them every-where in what follows and operate with the kernel of ~/aB as if it did not possess three-particle singularities at all. The proof of this assertion is rather long. 5. The two-particle singularities reflect the representation W~(P,P';z)=F~(P,P';z)+

G(P,p~';z)

qJd(k/) z--e,~--p,'2/2n,

+

~k z--~Z-f2/2n

~

G(p~,P';z)

na,n~

+ Zz ~,)--1

-- e~~-- p~2/2n~

I~dJ(P~'p/; z)

z -- e~~-- p/~/2n~

(2.5)

in which the f u n c t i o n s F, G, G, and J do not p o s s e s s t w o - p a r t i c l e s i n g u l a r i t i e s . E v e r y w h e r e off t h e p o s i t i v e and d i s c r e t e s p e c t r u m of H t h e y a r e b o u n d e d d i f f e r e n t i a b l e f u n c t i o n s . On t h e p o s i t i v e s p e c t r u m t h e y h a v e t h r e e - p a r t i c l e s i n g u l a r i t i e s w h i c h c a n b e r e a d i l y d e s c r i b e d by u s i n g f o r m u l a (2.4) f o r the k e r n e l s of T~RoTB, into w h i c h one m u s t s u b s t i t u t e the r e p r e s e n t a t i o n (2.1) f o r t h e k e r n e l s t ~ ( l ~ , k~; z). 6. W e now g i v e s o m e i n f o r m a t i o n a b o u t t h e S m a t r i x . S u p p o s e t h a t a s c a l a r p r o d u c t i s d e f i n e d on an n - d i m e n s i o n a l a r i t h m e t i c s p a c e R n of p o i n t s P = {Pl, P2, 9 9 - , Pn}. L e t t h e c o r r e s p o n d i n g q u a d r a t i c f o r m b e 7~ = X(P). C o n s i d e r t h e s e t S~ - 1 d e f i n e d b y t h e e q u a t i o n ~(P) = 1 and t h e e l e m e n t dw of a r e a on S~ - 1 w i t h r e s p e c t to X. L e t p x(tX) = CxtX~n , where the constant c x is chosen such that

dP=dp~Adp2A...Adp.=

1220

p~(~)d~Ad~,

,,~here La= )~(P).

L e t Sn - 1 be the H i l b e r t s p a c e of functions on L2(S n--I k ) that is s q u a r e s u m m a b l e with r e s p e c t to do~. T h e s p a c e c o r r e s p o n d i n g to the continuous s p e c t r u m of h can be r e p r e s e n t e d as a d i r e c t i n t e g r a l

@L2(Sho2)p~(E)dE, o

i . e . , its e l e m e n t s can be r e g a r d e d as functions f(w, E) on S 2

h 0, depending on E, w h e r e

11/11 ~

= j" [ll~P~(E)dcodE.

H e r e , h 0 is the f o r m k 2 / 2 m . In this r e p r e s e n t a t i o n the s c a t t e r i n g o p e r a t o r is given by the S m a t r i x s(E): L2(S~o ) ~ L2(S~o), which can be d e s c r i b e d by m e a n s of the k e r n e l 6(r

r

-- 2n~pho(E)t(E'/~co, E'/,r , E + iO).

(2.6)

i

S i m i l a r l y , the s p a c e of the continuous s p e c t r u m of H is r e a l i z e d as the d i r e c t i n t e g r a l

~. Z H e r e , h ~ is the f o r m p ~ / 2 n ~ . i n t e g r a l the h i g h e s t channel.

QL~(S~h~

OL~(S~~

We shall call the t e r m s in the f i r s t i n t e g r a l the l o w e r c h a n n e l s and the second

T h e S m a t r i x S(E) on the l o w e r c h a n n e l s a c t s as an i n t e g r a l o p e r a t o r with a m a t r i x k e r n e l of the f o r m S~(6o~,, co~'; E) = 6~5%(o~,, ~o~') -- 2ni9,, ~ (E -- e~)J~J((E -- e~')'/~o~, (E -- e~J)'/~o~'; E-if- i0).

(2.7)

T h e s e c o n d t e r m s in t h e s e k e r n e l s depend sm.oothly on. all the a r g u m e n t s . F o r p o s i t i v e e n e r g i e s the S m a t r i x S(E) contains the i n t e g r a l o p e r a t o r s S~a and S~o , which connect the l o w e r and h i g h e s t (denoted by the s u b s c r i p t 0) channels. We s h a l l not give e x p r e s s i o n s f o r t h e m but note only that the k e r n e l s of t h e s e o p e r a t o r s depend s m o o t h l y on all the a r g u m e n t s . T h e k e r n e l of the S m a t r i x Soo : L2(S~o ) ~ L2(S~o ) on the highest channel is singular: Soo(r

~)') -- 2~ip~o(E)T(E'/~o), E'/,oS; E + iO).

(o'; E) = 6(r

It can be s e e n that it includes all the t h r e e - p a r t i c l e 3.

Trace i.

(2.8)

s i n g u l a r i t i e s of the k e r n e l of T(z).

Formulas Consider the o p e r a t o r s

to(z) = r(z) - to(z), Rc (z) ---- R(z) - - R0(z) - - Z

(R~ (z) -- n0(z) ). r

T h e f i r s t is a n u c l e a r o p e r a t o r off the s p e c t r u m of h and the second is not n u c l e a r ,

t t o w e v e r [8]

d

dz Re(z) is a n u c l e a r o p e r a t o r off s p e c t r u m of H and this i m p l i e s n u c l e a r i t y of Im Re(z) = -~- [1/~ (z) -- R e ( g ) ] = 2~-~ d~ ~( -~ )R~ ,d z

~

w h e r e the i n t e g r a t i o n is a r o u n d a contour that s u r r o u n d s the s p e c t r u m of H. C o n s i d e r the t r a c e s ~(z)---~spImrc(z),

I m r o ( z ) = 2-~-[r~(z)--rc(~)] ,

'f~ (z) ---~Sp Im R~(z).

1221

One can s h o w t h a t t h e f o l l o w i n g l i m i t s e x i s t :

~•

a>0,

E>0,

a$o

e > e .... E~U{e~'}[J{O }.

~?•

L e t r be a h o l o m o r p h i e function d e f i n e d in the n e i g h b o r h o o d of t h e s p e c t r a of h and H t h a t d e c r e a s e s at i n f i n i t y a s an i n t e g r a l p o w e r of z -1. In t h e g r o u p i n t e g r a l ~0(z) = e -Bz to w i t h i n a s c a l a r f a c t o r . Set

b(h)]o = (~(h)

~(ho),

-

b ( n ) ]o = ~(U)-- ~(H0)--~, (~ (}L)-- ~ (~0)). These operators possess traces:

sp[q)(h)]~ = Z

q)(e{)j- ~ - i q)(E) ~+ (E)dE,

i=i

(3.1)

o

Sp[~(It)]~ = E ~ ( E ~ ) + - ~

J~(E)9~(E)dE.

(3.2)

m

T h e p r o o f of t h e s e a s s e r t i o n s f o r h can b e found in [3, 4]. a r e g i v e n i n [81; t h e y a r e m o r e c o m p l i c a t e d .

T h e p r o o f s of s i m i l a r r e s u l t s in the c a s e of H

2. T h e m a i n p r o b l e m h a s now b e e n r e d u c e d t o t h e e x p r e s s i o n of the l i m i t s w+(E) and ~+(E) in t e r m s of the S m a t r i c e s . W e s h a l l c a l l t h e c o r r e s p o n d i n g f o r m u l a s t r a c e f o r m u l a s . In t h i s s u b s e c t i o n w e s h a l l deduce certain preparatory identities. ' Consider the unitary operator s(z) =

h--~ h--z

h0--z - , ho--~-

z = E @ ie,

e =/: O.

We t r a n s f o r m t h e r i g h t s i d e ho--Z --= [e + 2ier(z) ] [e - - 2~ero(~) ] = [e + 2~er~(z) h--z ho--Z - - 2iero(z)t(z)ro(z)] [e - - 2iero(i-) ] = [e + 2iero(Z)] [e - - 2iero(5) ] h--~

- - 2iero(z)t(z)ro(z) [ e - 2iero(~) ] = e - - 2iero(z)t(z)ro(Z). Differentiating sp In s (z) = s p i n

h--5ho--z h--z ho--~ '

w e o b t a i n t h e m a i n p r e p a r a t o r y f o r m u l a f o r two p a r t i c l e s 2io)(z)=sps*(z)

Os (z) OE '

z ~---E + / e ,

(3.3)

where s(z) = e - - 2iero(z)t(z)r~(Z). S i m i l a r c a l c u l a t i o n s w i t h s l i g h t a d d i t i o n s l e a d to t h e m a i n p r e p a r a t o r y f o r m u l a f o r t h r e e p a r t i c l e s 2i~(z)----Sp(S*(z)~)

, c

where

1222

z=E+ie,

(3.4)

,,

0S(z) \

,

0S~ ~

s~ (~)

,

and S(z) = E

-

-

2isRo(z)T(z)Ro(Z),

S~ (z) = E - - 2ieR0 (z) T~ (z) Re (Z) are unitary operators. 3. W e now t r a n s f o r m t h e p r e p a r a t o r y f o r m u l a s to a f o r m in w h i c h a l l the d i f f e r e n t i a t i o n s a r e t r a n s f e r r e d to t h e k e r n e l s of t and T . U s i n g t h e c o n c e p t s i n t r o d u c e d in t h e l a s t s u b s e c t i o n of S e c t i o n 2, we s h a U d e s c r i b e the s t r u c t u r e of t h e d i f f e r e n t i a l o p e r a t o r s t h a t a c t on t h e s e k e r n e l s . L e t Q ( P , p ' ; z) b e a f u n c t i o n on R n • R n t h a t a l s o d e p e n d s on t h e p a r a m e t e r z, z E C, I m z ~ 0. L e t VkQ b e t h e i n t e g r a l o p e r a t o r w i t h k e r n e l ps-'()~(P'))

] 0

0~

0

(-~-~-~7+~--)p~(~(P'))Q(P,P';z),

z=E+ie.

H e r e , t h e d i f f e r e n t i a t i o n w i t h r e s p e c t to X = X(P) [~' = X(P')] i s f o r f i x e d P'{P) a n d z. With this notation 2i~o (z) --~ sp Kho(t),

(3.5)

where Kho(t) -~ - - 2ieroro* Vh~t - - (2ie) 2roro't'roro* Vhot, and 2i~ (z) = S p [ K , 0 ( T ) - - 2

Km(T~) ]'

(3.6)

where KZo(T) = - - 2ieRoRo" Vz0T - - (2ie) 2R3R0"T'RoRo" V~oT. We shall call these the preparatory formulas for single-channel scattering. (3.5) a n d (3.6) a r e i d e n t i c a l s . L e t u s v e r i f y (3.5).

E s s e n t i a l l y , t h e p r o o f s of

Note f i r s t t h a t (3.3) i m p l i e s 2i(o (z) = - - 2 i e - - ~ sp r0r0*t - - (2ie) sp r~ro t* --~roro t.

(3.7)

T o t h i s end we show t h a t 0 0 sp rot*ro* ----~rotro* - - sp roro*t* --~roro*t :

sp rot*ro*rotro* ~ - - sp roro*t*r0r0* ~t --~ 0.

Here, we have used troro't" ----t'roro't. L e t u s a n a l y z e t h e f i r s t t e r m of (3.7): 0 - - 2i~ - ~ s p

0 roro*t ---- - ~

dk

- - 2ie

(ho(k)--E)2+e 2

t(k, k; E -4- ie).

D i f f e r e n t i a t i n g the f i r s t f a c t o r w e go o v e r to d i f f e r e n t i a t i o n w i t h r e s p e c t to h 0 and i n t e g r a t e by p a r t s with r e s p e c t to h0, r e m e m b e r i n g t h a t dk = Ph0dwdh0 . T h i s t r a n s f o r m a t i o n i m m e d i a t e l y y i e l d s t h e f i r s t t e r m of (3.6). A s i m i l a r t r a n s f o r m a t i o n in the s e c o n d t e r m of (3.7) y i e l d s --(2ie)Zspr~176176176 V~~ A-(2ie)Zspr~176176

/ O__~t.) ror ,t" ( 00 ~ t ) --(2ie)~sP ro r / \0h~.

1223

Here, 3 t / 0 h 2 denotes differentiation of the kernel of t with r e s p e c t to h0(k) with r e s p e c t to the second a r g u ment. The t e r m s containing 0 / ~ h 2 can be r e p r e s e n t e d in the f o r m (2~e)~sproro * [ O--O---t*roro*t~ ~ ( 2 i e )

\ Oh~

I

2

t 0 **\ 9 sp ~--:--troro t }roro ,

( dt~2

I

which vanishes by virtue of (3.8). 4. A direct passage to the limit ~ ~ 0 in the p r e p a r a t o r y formula (3.5) for h leads immediately to the well-known relationship i

ds(E)

~(E) =-~sp~*(E) d ~

(see the introduction). Here, we use the smoothness of the kernel t(z) and the limiting relation 2ieroro* -+ 2ni6 (ho -- E). These arguments are inapplicable in the case of three particles since the kernel of T is now singular on the continuous spectrum. Moreover, as we have already mentioned in the introduction, the result obtained by the formal passage to the limit is devoid of meaning and requires regularization. In this subsection we shall assume that the pair Hamiltonians h a do not have bound states. Then the kernel of T contains only three-particle singularities. They are concentrated in the first iterations of Faddeev~s equations. We set

a

rt~

0

9

Here, T = T + T and the t h r e e - p a r t i c l e singularities of ~' can be neglected. We shall denote by S the sin0 gular part of the S m a t r i x which a r i s e s when T instead of T is substituted into (2.8). In this notation the r e s u l t of the passage to the limit ~ -~ 0 in the p r e p a r a t o r y formula (3.6) still has the f o r m 0

f]+(E)~---~-Sp S*~--dE

dE

-~ A(E),

where 1

A (E)~-- l i m - - S p [ K.0(T) -- 2

KH~

]

(3.9) 0 (see also [9]). However, this formula is not completely s a t i s f a c t o r y since although S*. dS~ and A(E) can be e x p r e s s e d in t e r m s of the c h a r a c t e r i s t i c s of the t w o - p a r t i c l e problems, these c h a r a c t e r i s t i c s cannot be reduced to S m a t r i c e s . More p r e c i s e l y , they contain t w o - p a r t i c l e T m a t r i c e s up to quadrilinear t e r m s . 5. Let us now continue the discussion of single-channel scattering. We shall show that the r e g u l a r i zation of Sp(S*- dS/dE)c can be p e r f o r m e d by means of the two-particle S m a t r i c e s . It is sufficient to v e r i 0 fy that the singularities of T on the energy surface can be expressed in t e r m s of the t w o - p a r t i c l e m a t r i c e s s a. The kernels of these singular t e r m s were written down explicitly in Section 2. In all the following formulas it should be noted that the a r g u m e n t s in the kernel of the S m a t r i x S(E) a r e connected by the equations H0(P) = H0(p') (they a r e situated on the t h r e e - p a r t i c l e energy surface). For T a we obtain k'2 ) T~(P,P';z) -~ T ~ ( P , P ' ; E - ~ - i O ) = t~ k . . .k' . . - -~ . -~iO 5 ( p ~ 2m~

p~),

and since k~ = t~2 (because Pa = P~) on the t h r e e - p a r t i c l e energy surface, t a with arguments in a c c o r d a n c e with (2.6) can be e x p r e s s e d in t e r m s of sa. Here, it should be noted that the t w o - p a r t i c l e S m a t r i x sa is constructed in a c c o r d a n c e with the f o r m ~ = k ~ - 2 m a and is r e l a t e d to Sa by the equation

Similarly, for the o p e r a t o r TaRoT s we obtain

1224

-i

, k~ ~ ~, iO) t~(k~, '~ 2m~ kn (T~RoT~) (P,P';z)--~t~ (k~, k~;~-~m

2m~

k[._i0] 2m~

.

(3.10)

Note that

__.-~. . . . 2rn~

2n~

2rn~

§

2n~ '

and, t h e r e f o r e , the singular factor in (3.10) can be r e p r e s e n t e d in the f o r m [ k~ 2ra~

kj ~ 2m~

i0] -~ .

It is clear that on the manifold on which singular factor vanishes the t w o - p a r t i c l e o p e r a t o r s ta and tt~ have arguments of the type

t~(E'!%,. E'!%);; E -{- iO),

t~(E'!~r E'l~o'; E § iO),

i.e., the singularities TaRoT ~ on the e n e r g y surface can also be e x p r e s s e d in t e r m s of sa and s s. T h e r e f o r e , instead of the singular operator S, one can use S, which is explicitly e x p r e s s e d in t e r m s of the t w o - p a r t i c l e S m a t r i c e s , to r e g u l a r i z e Sp(S*-dS/dE)c. The r e s u l t is

~+(~) = ~is ~ [ s ' ~

-- S*~dS+

~

(3.11)

(E),

where o

l

d$

A priori it is not clear whether A(E) can be expressed in terms of the two-particle S matrices. However, this is true and this assertion is the main result of the present paper. In addition, we shall give an explicit, albeit slightly modified expression for this operator later. 6. The product S* 9 dS/dE of singular operators also contains regular terms. The aforementioned modification consists of retaining only the truely singular terms in S*" dS/dE, combining the remaining 0 terms, for which the trace may meanfully be taken, in the quantity A. The outcome

is i

dS

9+(E) = - ~ - Sp [ (S" -~/_~ -- A (E) ] + 2~SP~I(E),

(3.12)

in which A and ,~ can be e x p r e s s e d explicitly in t e r m s of the two-particle S m a t r i c e s . The derivation of these e x p r e s s i o n s entails f a i r l y laborious and c u m b e r s o m e calculations, which we shall t h e r e f o r e omit and m e r e l y write down the final formulas. 7. As a p r e l i m i n a r y , we must write down a number of o p e r a t o r s that act on the space L2(S~0).,, We

(+)

(-)

have I, the identity o p e r a t o r ; Tas -~ (T~s + T a s ) / 2 , w h e r e the integral o p e r a t o r s T kernels p.0(E)

~(+~

i

r

E ' ~ -- E~ -- i0 t

p~ (E'~) p~(E'~)

I

) a r e defined by the

~)

G(o)~,o)~;E~)t~((o~,.(o~,

p~(E~) p~(E~)

(;

' I

E~ -- E'~ -- ~0

In these kernels sa = I~-2~it~, where ta is the t w o - p a r t i c l e T m a t r i x on the energy surface, the S m a t r i x s~, the coefficient p~, the arguments E~, and wa being constructed in accordance with the f o r m ~ = k~ /2m~. The arguments E'~, EBa, wa~, ' and w~a are defined in an exactly similar manner in terms of k~B and k ~ ; for example, E~B = k ~ J 2 r n ~ and w ~ = =a~ - ~B" In this notation, A(E) is given by

A (E) ~-- B(E) -~- 2~iC(E),

(3.13)

1225

where 9 dS~

dE'

d I'~] C = Z [ -- T~,*--~ S~S, + (S~ + S, -- I) *--dff 9 a=/=~

It is not too difficult to show that the operator (S*. d S / d E ) c - ~ has a principal kernel. It is m o r e difficult to calculate A. New notation is r e q u i r e d to d e s c r i b e this operator. In particular, it contains a s p e cial differentiation symbol 0. This differentiation, which is defined by its action on kernels, can be applied to sums of products of the kernels ta, the complex-conjugate kernels, and to integrals of the corresponding products. On a product, 0 acts as a differentiation, each factor of the type t~(wa, ~0a, " E~) being differentiated with r e s p e c t to Ea. This differentiation is applied to the r e m a i n i n g explicit f a c t o r s , even those that depend on Ea, as if they were constants. Applied to integrals, 0 is a s s u m e d to act in a c c o r d a n c e with the s a m e rule on the integrand. The composition O(d/dE) is to be understood as ( d / d E ) ~ - ~ t / E ) 3 ~ F u r t h e r , suppose that an o p e r a t o r Q(E) is defined by the kernel Q0~, where Q0 is a smooth kernel and ~ is a f~_~ction of one of the two kinds

,, =

l E ' ~ -- E~ -- i0

,~ =

6(~'o~ -

E~).

In this case, [Q]r is the o p e r a t o r defined by the kernel Q0. In this notation

2I = 4-~ O[B]T + ~i (-~-O-- tE dEd E) CR-- ~iCR,

,

(3.14)

d

,z=~p

This completes our description of A and A. 8. If the scattering is not perfectly particle singularities. In the interval eTa is facilitated by the fact that 2iCRoR ~ ~ 0 we m a y mention a technical detail. After namely

elastic, then the kernel of T p o s s e s s e s two- as well as t h r e e < E < 0 the p a s s a g e to the limit in the p r e p a r a t o r y f o r m u l a (3.6) and the absence of t h r e e - p a r t i c l e singularities in this ease. Here, the passage to the limit we wish to obtain the desired result,

/

~+(E)~--#SpS*(E)

dS (E____)_) dE , E
(3A5)

directly. This m e a n s t h a t the p r e p a r a t o r y formula must be r e a r r a n g e d somewhat. The r e s u l t of the r e a r r a n g e m e n t , the s o - c a l l e d p r e p a r a t o r y formula for the lower channels, is given in our note [9]. Without going into the details, we should like to emphasize once m o r e that the combination of the given f o r m u l a and the r e p r e s e n t a t i o n (2.5) in Section 2 leads readily to (3.15). It does not r e q u i r e regularization. The t r e a t m e n t of multi-channel s c a t t e r i n g for E > 0 r e q u i r e s the combination of the p r e p a r a t o r y f o r m u l a f o r single-channel s c a t t e r i n g and the aforementioned p r e p a r a t o r y f o r m u l a for the lower channels. In the passage to the limit ~ ~ 0, c a r e must be taken with the singularities that a r i s e f r o m expressions of the type 2ieR0R* and the two- and t h r e e - p a r t i c l e singularities of the kernel of T. As we mentioned in Section 2, the two- and t h r e e - p a r t i c l e singularities do not intersect; this simplifies the investigation and reduces it to a combination of the p r o c e d u r e s developed for single-channel s c a t t e r i n g and for the }ower channels in the case E < 0. The final r e s u l t is again a f o r m u l a of the type (3.12):

~+(E)=~Sp

[(

S* "h-k-, dS~ o - - A ] + 5V t sv;~' E > ~

(3.16)

in which the S m a t r i x S(E) participates fully. The additional t e r m s due to the lower channels a r e given by smooth kernels and do not r e q u i r e regularization. The o p e r a t o r s A(E) and ,~(E) a r e the same as in the single-channel case. 1226

9. In formula (3.16) one can make the passage to the limit in which the m a s s m 3 tends to infinity. If, in addition, vt2 = 0, one obtains a model with seperable variables. Its investigation r e d u c e s to a study of the t w o - p a r t i c l e problems, the following equation holding: d2 i dE'lndets2~(E ~) "lndets3i(E--E~). ~+(E)-- -~- i d-~

(3.17)

0

In this model S = $23 9$21; t h e r e f o r e , the f i r s t t e r m in (3.16) vanishes since it contains the logarithmic derivative of det SS~3S~1 r e g u l a r i z e d by separation of the "truly" singular t e r m s . Subsequent t r a n s f o r m a t i o n s of S p ~ ( E ) / 2 i lead to the right side of (3.17). LITERATURE

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

CITED

E. Bethe and G. E. Uhlenbeck, Physica, 4, 915 (1937). L . D . Landau and E. M. Lifshitz, Statistical Physics, London (1958). V . S . Buslaev, Dokl. AN SSSR, 143, 1067 (1962). V . S . Buslaev, in: P r o b l e m s of Mathematical Physics [in Russian] , No. 1, LGU (1966), p. 82. F . T . Smith, Phys. Rev., 131, 2803 (1963). F . A . Berezin, Dokl. AN SSSR, 157, 1069 (1964). F . A . Berezin, Proc. International Symposium on Many-Body P r o b l e m s , Novosibirsk, 1965 [in R u s sian], Nauka (1967). V . S . Buslaev and S. P. M e r k u r ' e v , Trudy Mat. In-ta AN SSSR, 110 (in press). V . S . Buslaev and S. P. M e r k u r ' e v , Dokl. AN SSSR, 189, 269 (1969). L.D. Faddeev, "Mathematical problems of the quantum theory of s c a t t e r i n g f o r a t h r e e - p a r t i c l e s y s t e m , " T r . Mat. In-ta AN SSSR, 69 (1963). R. Dashen, S . - k e n g Ma, a n d H . Bernstein, Phys. Rev., 187, 345 (1969)o R. Dashen and S . - k e n g Ma, J. Math. Phys., 11, 1136 (1970).

1227