On the three-body Coulomb scattering problem

ANNALS

OF PHYSICS

130,

395-426 (1980)

On the Three-Body

Coulomb

Scattering

Problem

S. P. MERKURIEV* Institut

fiir

Theoretische

Physik,

Freie

Universitiit

Berlin,

Amindee

3, Berlin

(West),

Germany

Received April 22, 1980

We describe the smoothness properties and the asymptotic form of the Green’s function (in configuration space) for three charged particles. We also discuss the integral equations and the boundary value problems for the Coulomb wavefunctions and we show that they form a complete set. Finally, we study the singularities of the Coulomb scattering operator, and we investigate the connection between the Dollard wave operators and the Coulomb wavefunctions.

The mathematical aspects of the scattering theory for a system of three neutral particles were first considered by Faddeev [l]. He studied the properties of the Green’s function and of the wavefunctions by means of integral equations in momentum space. The Faddeev equations, in both integral and differential form, are used for numerical calculations in nuclear physics (see, for example, [2, 31). Hence the scattering theory may now be considered to be in a satisfactory mathematical state. The situation was completely different for systems with a long-range interaction. It is well known that the Faddeev equations cannot be applied to the Coulomb scattering problem. Recently, however, progress has been achieved in solving the three-body Coulomb scattering problem. We shall describe here the principal results obtained in the last few years. The generalized Coulomb wave operators for few-body systems were defined by Doliard [4]. Different aspects of the non-stationary formulation of the scattering theory were studied in many papers (see, for example, (5-91). In particular, Vesselova [5] proved the Green’s function in momentum space to have the distorted pole singularities ($ _ z)-~+‘c, (p,” + 6, - +I’ ig,, (1) which follow from the existence of the Dollard’s wave operators. But the stationary formulation of the Coulomb scattering theory did not exist, and the construction of the resolvent kernel and of the wavefunctions was not known. Several forms of integral equations were proposed to study these problems. Noble [IO] formulated modified Faddeev equations for systems with Coulomb plus short* On leave from Department

of Theoretical Physics, University of Leningrad. UdSSR.

395 0003-4916/80/140395-32$05.00/O Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.

396

S. P. MERKURIEV

range interactions. But the input kernels of Noble’s equations cannot be expressed in terms of known quantities and have to be studied separately. For this purpose, it is necessary to solve a pure three-body Coulomb problem. In fact, Noble’s equations cannot be used to develop the stationary formulation of the Coulomb three-body scattering theory. The first important step in solving the Coulomb scattering problem was made by Vesselova. In Refs. [l 1, 121 modified Faddeev equations were formulated for a system of three charged particles. These equations are obtained from Faddeev’s equations by inverting an effective two-body Coulomb singular part of their integral kernels. Vesselova proved the resulting kernels to be compact below the three-particle threshold. But above the threshold, such equations cannot be used to study the wavefunctions and the resolvent kernels. Recently, Alt et al. [13, 141 used the renormalization procedure proposed in Ref. [ll] for studying the three-body Coulomb problem to obtain modified AGS quasi-particle equations [15]. The compactness of the resulting kernels below the threshold is the consequence of the results proved in [I 1, 121. But above the threshold, the two-particle renormalization procedure [ll, 161 is not sufficient to guarantee compactness. The new Coulomb peculiarities arise from the interplay of the pole in the free Green’s function and Coulomb singularities in the two-particle T-patrix. It has been proved in [17] that the (3 -+ 3) scattering amplitude has Coulomb singularities of the form 1pB - pi i-3-in even if only two particles are charged. These three-particle singularities are different from the two-particle ones. Their appearance reflects the non-compactness of the integral equations proposed in [ 11, 13, ,141 above the threeparticle threshold. A new form of Faddeev’s equations with compact kernels was proposed in papers by the author [17-221. These equations were formulated and investigated in configuration space, and following results were obtained: (1) Smoothness and the asymptotic properties of the resolvent kernel R(x, x’, z) m configuration space were established for all z, ) Im z I > 0. (2) The complete set of the Coulomb wave functions was found. In particular, the expansion theorem for this set was demonstrated. (3) The unitary Coulomb scattering operator was studied. The singularities of its kernels were specified. (4) The connection between stationary and non-stationary wave operators was established. Hence the stationary formulation of three-body scattering theory is now known. It should be pointed out that all results of Ref. [17-221 concerning the compactness of integral equations and the continuous spectrum of the three-body Hamiltonian are true independently of the charge distribution. Coulomb forces may be either attractive or repulsive, and only smoothness properties of wavefunctions depend on the kind of Coulomb forces involved.

397

THREE-BODY COULOMB SCATTERlNG

This paper is organized as follows: In the first section, we describe the behavior-u of resolvent kernels for the threebody Coulomb problem. The second section is devoted to the Coulomb wavefunctions; in particular, we state the different definitions of stationary wavefunctions. In Section 3 we formulate the expansion theorem in terms of wave operators, and we describe the connection between stationary and non-stationary wave operators. Finally, in Section 4 we construct the scattering operator and discuss the singularities of its kern& 1. RESOLVEN'T KERNELS A. DeJinitions We shall consider a system of three pairwise interacting charged spinless particles. The two-body potentials V,(x,) (CL= 1, 2, 3) will be assumed to have the form of the sum of the Coulomb part and of a nuclear short-range one, ~‘L.kJ = =/ ,y, 1 + F$‘(xJ,

x, E R'.

The nuclear potentials 1’, satisfy the usual short-range conditions (see, for example, 131) They are smooth functions if / x, ) > 0. In the three-particle center-of-mass system we take as independent coordinates the usual pairs of three-vectors {xu , yJ (a = 1, 2, 3). Here, for example x1 = )‘I =

2m,m,

II2

(tn, + m3 1

(r2 - rd,

2m,(m, + m,)l’z (

ml

+

M2

+

fl73

m2r2+ w3ra I(

m,

+

n13

--r,

1

,

where ri (i = I, 2, 3) are particle coordinates and mi (i = 1, 2, 3) are their masses. We write {k, , p,} for the independent momenta which are conjugate to the variables (x, , yu}. We shall frequently combine pairs of coordinates (xn, yJ and momenta {k, , p,} into six-vectors X = x, @ yrr, P = k, @ P, . The transition from one pair {x, , y&j to other one {x6. yBj, N f p, is given by a rotation in R6 -yi3= GaX, + &,Y,

Yo = -&3,x,

+ GYn

ctTw+s;,=

3

>

1,

e,fl = 1,2,3,

where the coefficients C,, , S,, can be expressed in terms of the particle masses r7zi[3]. Finally, we denote by 4 a unit vectors, 4 = I q l-lq, q E R”. The three-particle Hamiltonian I-I is given by the expression

398

S. P.

MERKURIEV

where Oz,(dy,> is the Laplace operator acting on the variables x,( JJJ only. The Hamiltonians for the two-particle subsystems will be denoted by h, , a = 1, 2, 3,

These Hamiltonians will be assumed to have the bound states ~~,~(x,) with energies > 0 (i = 1, 2,..., IV,). If the Coulomb force is repulsive, 12, > 0 (attractive, -G,i 7 •~,i n, < 0), the number IV, of bound states is finite (infinite). B. Integral Equations Let J&(v), 0 < v < +, be the domain in the configuration space Rc6, where the distance 1.x~ / between the particles of the pair 01is much less than the distance 1y, 1 between the third particle and the a-pair centre-of-mass, when 1y, 1 + a: G,(V) = {X: 1x, 1 < a,(1 + / y. I)“, a, some constant}. Let Q,,(v) be the domain where the complementary inequality is true, and Q0 = nol Q,,, . Let x.(X, V) be a smooth finite function which equals 1 in Q(v) and 0 in Qm(vl), v < Vl < 4. We shall separate the two-body potential V, into two parts: VJXJ = P&Y> + V,‘“‘(X). Here, pa(X) is the “short-range” p a =m+nu OL Let Ho be the “asymptotic”

term and VL”) is the “three-particle

term,

y(o) = (1 - x ) n, a OL14

I x, 1Xa ’ Hamiltonian

long-range”

generated by the “long-range”

parts,

Ho = Hk + c Vf).

Here Hk = --d, is the free three-particle sum H, = Ho + pm. Let R,, be the resolvent components

Hamiltonian.

&a(z) = Ro(z)(&~~s - ~Az)

&)

We shall denote by H, the

R,(z),

where Ro(z), R(z) are the resolvent operators, R(z) = (H - z)-l,

R,(z) = (Ho - z)-‘.

The components R,,(z) satisfy the modified Faddeev equations

R& = (R,(z) - &ONb - c &x(z) ~&s(z),

(2)

THREE-BODY

COULOMB

399

SCATTERING

where the operators R,(z) and R,(z) have to be given. But in contrast to the case of neutral particles, there are no explicit expressions for the kernels R,(X, X’, z) and R,(X, X’, z). These kernels are to be studied with the aid of independent equations of the Lippmann-Schwinger type.

To obtain such equations we have developed a new method. Its principal aspect is the construction of asymptotic solutions of the various SchrSdinger equations using the locality of the differential equations involved. For example, we can construct explicitly the kernel R&X, X’, z) satisfying (-A,,

- z) Ras(X, X’, z) = 6(X - X’) - (RasV’&(X,

‘- c e’(x~> ?

X’, z),

(3) R&X,

X’, z) = R&X’,

X, F),

where the term (Rasp&(X, X’, z) decreases fast enough if I X’ / + co, uniformly in X and z. This kernel may be taken as a zero-order approximation of the resolvent kernel R,(X, X’, z). With the aid of Green’s theorem, one can obtain an integral equation of the second kind for the kernel R,(z), namely,

R,(z) = Ras(4 - &s(z) V,(z) R,(z).

(4)

This Lippmann-Schwinger-type equation may be studied by well-known configuration space methods [23]. The compactness of the operator (R,V& may be proved for n> 1. Similar equations are true for the operators R, (a = 1, 2, 3). The smoothness and the asymptotic form of the kernels R, may be studied. It should be noted that Eq. (4) is analogous to the modified Lippmann-Schwinger equation used for studying the two-body scattering problem with pure Coulomb and nuclear short-range potentials [24]. The operator Ras plays the role of the pure Coulomb Green’s function [25]. Knowing the properties of the kernels R,,(X, x’, z) and R,(X, X’, z), one can investigate the mathematical aspects of the Coulomb scattering theory. All the details about the resolvent R(z) may be obtained from Eq. (2) by means of familiar Banach space techniques. C. Approximative

Kernels

In this subsection, we describe the asymptotic resolvent kernel Ras . First, we shall explain the construction of the eikonal approximation, which is the basis for studying the Coulomb scattering problem in configuration space. Let L be a solution of the eikonal equation IVL(2=

and let & = VL be the asymptotic

direction

1

of the motion

of particles in con-

GENERALIZATION

OF

THE

NON-LINEAR

o-MODEL

471

vacuum configurations n(x), maps R4 into a one-dimensional curve, rather than into a single point in S4. Finally we note that the dynamics of the O(5)-model can be expressed in terms of a quaternionic function W(X). Here W(X) is obtained from 11(x) by a stereographic projection of S4 onto R4 N H, i.e., W(x) ==

introducing

n,(x)

L in,(s) -1.,jna(r) 1 - n,(s)

W(x) into the Lagrangian ~

: 4 (a,w*

- /in,(s)

(32) it can be rearranged as

@M/)2 - (t!,w* i;,,w)(i“w’* ---. (1 pi- / M’/“)”

--

?“I+.) -

1401

In fact, this quaternionic representation of the O(5)-model gives a direct connection between this model and the HP-model, as we will discuss. Next, we briefly consider the complex CP2-model. (Apparently this model is not equivalent with the O(5)-model.) The dynamical variable of this model is a complex unit vector Z(X) in C3. In the same way as for the two-dimensional CP-models, a U(l)-gauge symmetry is involved here. It corresponds to phase transformations of the vector z. The order parameter of the model is therefore obtained from the unit sphere in C3 (z+z = 1) by dividing out the action of this gauge group; consequently it takes on values in the complex manifold CP2. Although CP2 is a closed manifold of four real dimensions it is not a four-sphere S4. In fact the fourth homotopy group of CP” is trivial. 7r4(CP2) = 0, thus showing that the topology of the CP2-model is quite different from that of the O(5)-model. The vector spaces .9(x) in this model are two-dimensional complex spaces, and the gauge group of the associated vector bundle is therefore U(2). The corresponding gauge field is given by F = V+Dz A (Dz) t I/‘. (41) The action of the CP model is defined in the same way as for the O(5)-model, and we therefore have S ::- ; j 11F j/2 d4s (42)

== ; .i 11Dz A (Dz)+ !I2 #I. Comparing this with the action of the O(5)-model we see that the only difference is that ordinary derivatives here have been exchanged with covariant derivatives. This is related to the fact that there is now a non-trivial gauge group associated with the FL-bundle. We finally consider the HP-model. This model is closely related to a model previously proposed by Lukierski [3] in the context of more general HP-models. The

THREE-BODY

COULOMB

SCATTERING

401

The values fL , A, and W, are defined by

The quasi-potential

pa, satisfies the inequality

in this case. On the other hand the phase shift W, becomes infinite if the conditions (8) are not satisfied. The parabolic equation method [27] was used for constructing the asymptotic solution in such singular direction. As a result, the function Y,, was expressed in terms of certain special functions, and the quasi-potential VaSsatisfies the inequality 6 > 0, (11) I &&K P)I < a1 f I x l)Pd7 in this case. Y-‘&X, P) is a smooth function of the parabolic coordinates eCa)and 5, 5 = I x I - @, a. Finally we shall describe the kernel R&X, X’, 2). We require Ras(X, X’. ;) to be equal the free three-particle Green’s function R,,(X, x’, 2) = R&c,

,k I, Z) = - (W2

H3(1)(z1/2 I x - X’ I) / x - X”2

if the distance between the points X and X’ is sufficiently small, I X - X’ 1 < (1 : / X /“, v < 4. If the difference ( X - X’ ) tends to infinity, / X - X’ \ ---f ~3, the kernel Ras is defined with the aid of an approximative “distorted plane wave spectral integral” (27r-6 p

xs(P’,

E)

Y&Y,

P’) YJa3(X’, P’) p’2 - z

(12)

Here, xS is a smooth cutoff function, equal to 1 if P’” = E, E --1 Re z, and equal to 0 if PI2 > max{l, E + 1). Asymptotically, the kernel Ras may be represented in the form R&X,

X’, z) = C,, / X - X’ I-5/z exp {i(z)

I X - X’ I: C&,(X, X’, z).

C,, = e(~~/4)~z3/4(8 dz $P)--1.

(13)

The function In &, is given by eikonal formula (6) with I.. = j X - X’ I in almost al] directions of the configuration space. For example, the principal term of Gas is equal to

if X and X’ lie in Q, . The quasi-potential p&X, X’, z) satisfies ,the inequalities (10) (1 I), where the vector P is to be replaced by X - X’.

402

S. P. MERKURIJZV

Similar equations may also be obtained for the operators R, . The zero-order term R& for the resolvent kernel Rw is given by formulas (12) (13), where the function ??$ is equal to the sum Y= + Vu) and the term U(O) is generated by the single scattering eikonal 2, = ( K, 1 I x, I + (pm , v~). This function was described in [ 171. (See also below.) The integral equations for R, are of the Fredholm type R, = R&i + Rtl”,‘~ik’%, ,

where the operator (R&‘pg’)n,

(14)

n > 1, is compact in the appropriate

Banach space.

D. Asymptotic Form of Resolvent Kernels We shall define a class of functions in terms of which the construction functions and of wavefunctions will be described. Let Q.,J y, , I&J be the distorted spherical wave in R3

Qb.i = I Y,

I-‘exp

CZk’i” I Y, I + iK,J,

of Green’s

(15)

and Q,, be the distorted spherical wave in R” Qo(X, E) = 1Xj-5/2exp W,(X) = ; ng

[iE”‘I

01

XI + iW,(X) + iti,

/X/ln2E1/2jX1.

(16)

Consider a set r?(a) of functions f(X) which are twice differentiable if 1x, 1 > 0, 01= 1, 2, 3, and are of Holder class if 1x, ) = 0, 01= 1, 2, 3. We denote by yE(@), q E Rn, n = 2, 3,..., the class of smooth bounded functions f(x, 4, E), f o @), that may be represented as the sum 3 f(K

4

E>

=

Nfi

1 c &l&=1

Its,i(-d

Q(YB

, 4

4~)

+

G@'-,

4, El,

(17)

where the terms U,,! and U,, have a fixed asymptotic form if 1X 1-+ co. The functions UB,j are the distorted spherical waves (15) where EB,j = E - E~,$, and U, is the distorted spherical wave (16) U&j - As.j(.?B , 4, E> QOJ(YR7 ELT.A uo - A,(% 4 E> Q&K E). The amplitudes A&j$, 4, EBsi) and A,(-%‘, $, E) are assumed to be sufficiently smooth bounded functions. We denote by yE(*I the class of smooth bounded functions

THREE-BODY

COULOMB SCATTERING

403

whose asymptotics (17) contains the terms #,,iCJs,< corresponding to one index N only. The class vE(O)does not contain any such term. When attractive Coulomb forces exist, we shall admit the amplitudes A,(& 4, E) to have square integrable singularities with respect to 8 and 4 on the unit spheres i 2 j =: 1, 1q / = 1. Note that in this case, the functionsf(x, q) and r/,,?( y, 4) have square integrable singularities with respect to 4. It should be noted that the classes 9, , &’ and q$) can all be given the structure of a Banach space (see [I 9, 21 I). Let I?, be the complex plane C2 with the interval [E, cc) removed. Let wF , w,, , w, be the set of points in 17, where the homogeneous equations (2), (4), (14) have nontrivial solutions respectively. We also introduce neighbourhoods D(w~), D(w,). D(w,) and

f E 02)

Put 17: = rr,\D’,

D’ = D(wo) u (u NW,)) u D(wF). i

From the integral equations (2), (4) and (14), which have compact kernels in suitable Banach spaces, the following assertions may be derived: THEOREM 1. The Green’sfunction R,(X, X’, 2) belongsto the class yi’)(x’)( boo)) if X’(x) is fixed and ( X - X’ j Z 6 > 0 for z ~lIl~\~D(w~). The limits IimrL,, R(X. X’. E k ie) exist if E 4 D(o,), and the asymptotic representation

R,(X, A”, E & i0) - Ap”*‘(X,

P) 0,(X’, E)

(1X)

3 true I@” ! X j < C( 1 + 1X’ j)y, v < &, p = - E1lZx’, i X’ j + cc. THEOREM 2. The Green’sfunction R,(X, A”, 2) (LY= I, 2, 3) belongs to the class y%‘(X)( &‘(X’)) if X’(X) is jixed and / X - X’ I b 6 > 0, Z EIL,~\(~(uJ,J u where E, = max, E,,~ . The limits lim,l, R,(X, A”, E f k) exist if (UCYI> E $ w,, u (U,, w,). The asymptotic formulas

R,(X. A”, E + i0) - A;$*‘(X,

P) Q,(Y.

E)

THEOREM 3. The resolvent kernel R(X, X’, Z) belongsto the class p)JX’)( yz(X)) if X’(X) is jxed and 1X - X’ ) 3 6 > 0, z E II:, , ,llhere E = max, E, . The limits lim R(X. X’, E f ie) exist ifE $ D’. The asymptotic representation

R(X, X’, E i i0) - &)(X, 3

P) Qo(X’, E) Na (18”)

is true if

i X j <

C(l

+

I X’

I)“,

v <

&, P =T

-E’/2L?‘. P, zz --?I j X’ j-IElP.

404

S. P. MERKURIEV

It should be noted that the slowly decreasing spherical wave terms in the asymptotics (18”) of the Green’s function R(X, X’, z) generate the singularities (1) in momentum space. We have described the asymptotic form of the Green’s functions only for the case where one point X (or X’) lies much farther from the origin than the other one. The general case is considered in papers of the author [20, 221 and we do not give the corresponding formulas here. It should be pointed out that the resolvent kernels R&Y, X’, z), Rt(X, X’, z) and R(X, X’, z) satisfy inhomogeneous Schrtidinger equations because they are smooth functions if X # x’. For example, the resoIvent kernel R(X, X’, z) satisfies the equation -A,

+

(

1 v,(x,) m

-

z

R(X,

x’,

z) =

s(x

-

1

xl>

(1%

and a similar equation with respect to X’.

2. WAVEFUNCTIONS A. Definition of Wavefunctions Inserting the asymptotic representation (18”) into equation (19) we may conclude that the amplitudes &i(X) and #j(X) satisfy the homogeneous SchrSdinger equation (-Ax

+ C v,(x,) - E) A(+)(X) = o a

with respect to the X variable. We define normalized tonian H by the relations Y&c, P) = -2 ( P p(27ryi2

wavefunctions for the Hamil-

A,(X, P), (20)

%,i(X, Pa> = 2(2~)“‘” Aa,dX, pa>. Normalized wavefunctions YiO), Y,j@ and !Pi=l for the asymptotic Hamiltonians Ho and H, are defined similarly. The indices (3t) are omitted here. It should be noted that the three-body Hamiltonian wavefunctions describe the physical situation completely. The wavefunctions Y’a,i(X, P,) describe the scarttering processes (2 + 2) and (2 -+ 3) with two clusters in the initial state; the pair 01is in the bound state &JxJ. The wavefunctions Yo(X, P) correspond to the scattering processes (3 + 3) and (3 + 2) with three free particles in the initial state. Apart from the formulation given above, the wavefunctions may also be defined either with the aid of certain integral equations or in terms of certain differential equations together with asymptotic boundary conditions.

THREE-BODY

For example. the wavefunctions

COULOMB

!Pmand YE,i may be represented

a = 0, (CL,i)-,

Yy, = c q$,

405

SCATTERING

a = I. 2, 3; i

as the sum

= 1,2 ,..., N, .

R

Thus a is an index for the possiblechannels. The components YAB)satisfy the modified Faddeev integral equations 7p

a

=

qj Xn nR -

zR R,j(E + i0) &p!t

(21)

where the non-homogeneous terms x$’ are given by 6&Xl7(0) = Y’$Y 7 P) 6RI ’

a=0

(B) 3 y/(4 6aaxa = cd3‘x.13

u = s.

The components Yi8) are related to the sumsx:a R,, by equations similar to (18”), too. The wavefunctions Y,$“) may be defined as solutions of the integral equations (4) where the free term is to be replaced by the asymptotic solution Yas . Analogous equations are true for the wavefunctions YF) and Yi:j . B. .4syn1ptotic

Form of Wavefunctions

To investigate the differential formulation of the Coulomb scattering problem, we have to know the asymptotic form of the wavefunctions. This asymptotic behaviour may be studied with the aid of integra1 equations, or else direct asymptotic methods developed in papers of the author [ 18, 191may be used. We shall simply state results. The wavefunctions Y/,(X, P,) may be represented as the sum Y’,( Y, P,,) = ~?LK P,,) $- f,,(X, P,),

(22)

where the term U,, belongs to the class ~)~,(p,,). Here. P,, = P, E, == P2 if a 0. and E,, P," - E,,,, P,, = P, if a = {OL, ij. The function fr) contains the slowly decreasing terms in the asymptotics of the state Y, , which corresponds to the initial state of the system. We shall denote by Ab,, the amplitude of the distorted spherical wave Qb(Th , P,) appearing on the righthand side of (22). Here, X, = X if b = 0 and X, =: J’,~if b = (&j>. The function fy’ is given by

for a = {ol, ij, and q,,$( y,,P,) is the two-particle wavefunction for the Hamiltonian h”,,i = -A,, + p%‘,.(y,) with the potential

406

S. P. MERKURIEV

If the quantity n,, (see (15)) is not equal to zero, the potential r=,J y,) is a long-range one, pa,i - n,,l y, j--l + O([ y, j-3. Note that the two-body Hamiltonians ABSihave a simple physical meaning. They describe the scattering of the third particle on a bound pair (Y,which generates an effective two-body potential rm\,.,(y.). If a two-body parabolic coordinate 5-@), &J = 1y, 1 - (pa , y,), tends to infinity, f”) 5 -+ co, the asymptotics of the wavefunction may be represented by the sum

where xJ’$ is the distorted two-particle plane wave X$ = exp {ih

, YJ + iwYl(l

+ O(@“‘)-31,

and

The function U$ is the distorted spherical wave with a singular amplitude

The principal singularity

singularities

of A$‘{ have the form of a pure Coulomb AZ{ = /&I

two-particle

+ 0(&J),

where ALo) is the Coulomb amplitude

/#o) a =

-m

exp {--iii,,

&ml

with cos 0, = (FE , PJ, q& = arg r(l

In sin2 (0,/2) + 2i7,) 9 sin2 (ti,/Z)

(23)

+ in,,), nao:= 442 1P, I. The sum

A”. cxt.at .=/+“!+A. L2.z

01z.oLa

may be interpreted as the elastic (2 + 2) scattering amplitude. Hence this amplitude has a non-integrable singularity due to the long range interaction. The rearrangement (2 -+ 2) amplitudes Aer,&&, P,), {pj} # {ai>, are smooth bounded functions. If all Coulomb forces are repulsive, the same holds for the breakup amplitude A,,Jx, Pa). If the Coulomb force between the particles of the pair /I is attractive, the breakup amplitudes A,,,&x, P) have singularities of the form (I x, 1-l 1X ()l/2. This singularity is square integrable on the hypersphere I X I = 1. We shall briefly discuss the principal consequences of the representation (22) for the wavefunctions Y,,,(X, PJ. First, note that the domain Q,(V) only the terms jtfj and &&J Uols,J ya, Pa) give non-trivial contributions to the asymptotics of the wavefunction Y&J’, Pa). The term V,,,JX, Pa) has order O(] X ]-5/z) (O(l X j-2))

THREE-BODY

COULOMB

SCATTERING

407

if the Coulomb force 12,] x, 1-l is repulsive (attractive) and does not give a contribution in the order O(\ X1-r). In the domain Q,(V), the term UoSaihas order O(/ X I-“) xa (1 + I x, j)-lj2 if E > 0 (above the threshold), and this is the principal asymptotic term. (The asymptotic form of the wavefunctions Y,,$ in the domain Q, was first described by Peterkop [28]). If E < 0 (below the threshold), the term UO,a:iis exponentially small in all directions of configuration space. Then the asymptotic form of ‘u,,i(X, P,) is defined by the terms j$yi , $o,iU,i,sj corresponding to opened channels {pj]. IfXiiQ0(/3 = 1,2,3),theterm UD,aiis no longer a distorted spherical wave. Rather, it may be represented in the form

where the function o,, is a smooth bounded function of xs and y8 and not only of the angle variables 2). The explicit formula for o0 was given in a paper of the author [IS]. Note that this function satisfies the inequality

where the function C(x/3) is bounded if nB < 0 and decreases exponentially when j x0 j j X 1-l - 0 if n6 > 0. It should be pointed out, however, that in order to define the wavefunctions ??‘m,i uniquely in the differential formulation, it is not necessary to know the exact form of Vo,ai in !& (/3 = 1, 2, 3). Only the inequalities (24’) is to be satisfied “a priori.” Next, consider the asymptotic behaviour of the (3 ---f 3) wavefunctions Y,,(X, P). The asymptotics of Y,,(X, P) may be described in terms of eikonal approximations involving four types of eikonals. We have described two of them above: plane eikonal (p, X) and spherical eikonal / A’/. Let us describe the other two, 2, , Z,, , which correspond to the single and double two-particle scattering, respectively. Let Z, be a single scattering eikonal given by

This eikonal corresponds to the process where the two particles of the pair 01interact and the third one moves freely (see Fig. 1). (a FIGURE

1

To describe the double scattering eikonal, we have to introduce some notation. Let (kaB , pap} be the Jacobi momenta of the particles after the double two-particle collisions /3 - (Y(see Fig. 2).

S. P.

MERKURIEV

FIGURE

The final momenta

2

ka4 and Pas may be expressed in terms of the initial

momenta

k, , Ps as follows:

where the intermediate

momenta qoa are given by

qoa = I kB I(cos O,,$, + sin Oolscos qosef’ + sin 8,, sin TaBef)),

and PO , e;“, eL3’ are the basic hectors (2) = -cot &j$ + csc 8,&, el3

ej’ = [eF’,$aJ,

cos 0, = (& ) 6,). The angles O,, and cpaacan be defined from the principle of least action

The single-scattering eikonal approximation

is given by

!Pm = C,(M,) 1x, 1-l exp{i(E)l12 Z, + SK}. Here, the Coulomb

phase shift W, has the form of the sum

where W’“’ 2;k ,ln2lk,Ilx,l R = - o! rY

ln( I kk) I I x5 I + (kp’, Ed),

(25)

THREE-BODY

COULOMB

409

SCATTERING

if /3 # CL The amplitude C,(M,) is an arbitrary smooth function of the “tangent” variable M, = X - 2, VZ, corresponding to the surface 2, = const. The local coordinates 2, , w, and u, may be used on the surface Z, = const: I -ycY 09

The double-scattering

eikonal

approximation

p,, may be represented in the form

qmR = (1 x, I I Y, 0-l GR&%~ expWi2& where the amplitude

+ i WE&,

(26)

A,, is given by

A& = (*)1’2

! 3;

/-1’2

cose, = (j, ,SJ.

The amplitude C,,3 is an arbitrary smooth function of the local coordinates L, , 0ao,g)orBandlMd,i~,,l = (X’ - Zi,J1/2 on the surface Z,, = const. The Coulomb phase shift W,, has the form of the sum w,, = c wg

(27)

where w”’aa = - a

2lL

In 2 i kaB 11X, !

if,j = 01and

ifj f 01. Now we may describe the asymptotic form of the wavefunction Y&X, P). The slowly decreasing term go(‘) from the right-hand side of (22) may be represented as the sum

where x0 is the distorted plane wave (9) if 1k, 1f(N) -+ co for (Y= 1, 2, 3. The other terms are due to the eikonal approximations pa and pa, . In “non-singular directions”, the terms U, are given by the eikonal formulas (25). The amplitudes C,(M,) may be represented in the form

410

S. P.

MERKURIEV

where fa is the two-particle scattering amplitude for the Hamiltonian h, . The function 8 W, is an additional Coulomb phase shift which takes into account the long-range interaction before the short-range a-pair collision. This function 6 W, is given by the sum

(29)

The variables [:I and c:’ are defined by the relations

The amblitudes

C,,(M,,)

of the double-scattering

eikonal approximations

are given by

Here fa and fB are the two-particle scattering amplitudes for the Hamiltonians h, and h, . The additional phase shift 3 W,, may be expressed in terms of the “angle” variables I A& 1, 4, , 0,, and vrrs with the aid of formulas analogous to (29), for which we refer the reader to [17]. It should be pointed out that the eikonal formulas are valid if all the amplitudes and phases are smooth bounded functions. However, there are many “singular directions” in configuration space where they became infinite. In these directions, the asymptotic form of # is described in terms of certain special functions (see [17, 221). As we shall see below, the “singular directions,” where one of the eikonals 2 = (p, X), 2, and Z,, coincides with the spherical eikonal 1X 1, play an important role in the behaviour of the scattering amplitudes. Let 5, & and tzB be the parabolic coordinates 4 = I x I - CR 9, and define neighborhoods

5,=1x1--zx,

&e3 = I Jr I - za, 3

V, , V, and V,, of the singular directions

vo = {X: 6 -=I (1 + I x I”), v. = {X: Lx < (1 + I XI% v,, = {X: Ll < (1 + I -A!I”>, where v > 0 is some constant.

THREE-BODY

COULOMB

SCATTERING

411

The function U, is a smooth function of the parabolic coordinate in V, . It has an integral representation involving the confluent hypergeometric function ~(a, c, c) [31] (see [21]). If 5 + co, the asymptotic form of U, is a sum of the distorted plane wave (9) and of the distorted spherical wave (16) with a strong singular amplitude A:@, P) given by A(y(X, P) = (1 - (P,

Jl))--BIE+iQ,I

A@‘, P),

(31)

where A(z, P) is a sufficiently smooth function and q. = xm (2 1k, 1)-l n, / p /. In V, , the asymptotic form of U, is described in terms of confluent hypergeometric functions ~(a, c, f,) [17]. If .$, --f 00, the term U, takes the form of the sum of the eikonal approximation (25) and of the distorted spherical wave with a singular amplitude Ai”) given by

where 0:) = (1 - 1X 1-l Za)1/2 and I?’ The value a, is defined by

is a smooth bounded function (see [17]).

The asymptotic form of the term U,, depends on the direction x in configuration space. Let Vi,“< Vi;)) be the domain where the inequality wEB> 0 (w,~ < 0) is satisfied, w al3 = E-1/2(l k,, j (pb0 , y,) - I PaB I I x, I). The quantity w,~ is equal to zero in the singular direction VoB. Due to the conservation of energy and momentum for classical particles undergoing double two-body collisions (/3 ---f u), I’,(l) is the allowed domain (asymptotically) and Vi;) is the forbidden one. In P’ks+‘,the term U,, has the eikonal form (26). In I’,, , however, it changes its form, and in Vi’,), it becomes the distorted spherical wave with a singular amplitude A$!J given by (34)

Here, &jJ is a bounded smooth function. The quantity aUBis expressed in terms of kinematic variables with a formula similar to (33) (see [17]). The vectors kLB and k,, are

In the intermediate region V,, the exchange behaviour of the term U,, is described in terms of the irregular confluent hypergeometric function #(a, c, taa) (see [17]). Note that the analogous exchange effect also exists in the case of neutral particles, where FresnelTs integral appears instead of hypergeometric functions [32]. In the domains Q,(V) (a = 1, 2, 3) the behaviour of the distorted spherical wave terms U,(X, P) in

412

S. P. MERKURIEV

(22) is given by relations similar to (24). For attractive Coulomb forces, however, additional singularities in the momentum variables 1” appear. They have the form (I I-J I I km /-‘Y2. The consequences of the asymptotic formulas of YJX, P) are analogous to the ones of the asymptotic formulas for YU,i(X, pJ which were discussed above. This completes our analysis of the asymptotic form for the wavefunctions Y$‘(X, pa) and Y,$+)(X, P). (The asymptotic form for the !P-1(X, P,) follows from YL-‘(X, P,) Y/(+)(X, -P,). We now formulate the boundary value problems for these wavefukctions. The wavefunctions YaSi(X, p,) and Y&X, P) may be defined as smooth solutions of the Schrijdinger equation

having the asymptotic form (22). In that representation the slowly decreasing terms -g8) have to be fixed according to the prescription given above, while the functions Xa of the class v,(P,) are then specified in terms of the solution. Alternatively, the wavefunctions Y&X, PJ and Y,(X, P) may be defined as smooth solutions of the modified Faddeev differential equations. Let

where the terms U, and UEj were described above, and

We shall denote by d,(Pol), a = 0, (a, i> the class of smooth vector-valued functions F = {f:‘), fL2), fA3)} whose componentsfL6) have the form

d*?,‘ =xoli,cix Pa) k,+ mx PA fP’(X, P)=x,‘:‘(x, P)+ Ud”‘(X, 0, where the functions UJ@)(X, Pa) belong to the class vf)(pJ. THEOREM

4.

(36)

Then we have

The modjied Faddeev equations in differential form (37)

have a unique solution in the class rp=(pJ. The sum C,, qg’ coincides with the wavefunction Y,(X, P,).

THREE-BODY

COULOMB

SCATTERING

413

The demonstration of this theorem is analogous to the one for neutral particles investigated in Ref. [3]. It should be noted that the Noble equations in differential form may be helpful for numerical calculations. These equations read

and may be proved to have a unique solution in the class @,,(P,) if the Coulomb forces between all particles are repulsive. We want to stress that Theorem 4 is true even if some of the Coulomb forces are attractive. The wavefunctions YA’)(X, P,) (a = 0, {G), oi = 1, 2, 3, i =~ I, 2 ... NJ defined above may be proved to satisfy the following completeness relations (in the tiilbert space Ls (P)):

(2x)--l 1 J‘IP,, !ry’( 0

x, P,) Y/y( Y’, P,,) =7 8(X - X’) --- P,,(,Y, X’).

Here Pd is the orthogonal projector onto the bound-state subspace for H. The next section is devoted to a proof of these completeness relations.

3. A. Stationary

WAVE

OPERATORS

Waue Operators

Let ju;, be the Hilbert space L2 (P) and ZU,( , u r-z 1, 2, 3, i --: I, 2 .. . .. n,, . be the Hilbert spaces L?(P). We shall denote by J? the orthogonal sum

Elements of the space PO will be denoted by f,(P) and elements of 2a,i by .f,., . The elements of .X? are vector-valued functions f, j =~ {.f, , ,lj.J. pi = I. 2. 3. i I, 2 ... N,, Let 8, be the reducible operator in 2 defined by

Here. l?” and g*,, are multiplication

operators

acting as

414

S. P. MERKURIEV

Consider the wavefunctions (27r)-3 Y&Y, P) ((27r)-V&X, pJ) as the kernels of operators UO(U,,i) from #O(&Si) to % = L,(R6). In the mixed X - P representation, these kernels are q*yx,

P) = (25-)-3 !P;*yx,

u:;tx

Pa> = cws

P),

lu,(:yx Pd.

Let U(*) be an operator from Z? to s%’ which acts on the elements of S@as Pf

= u,"lr,

f 1 u,'yjY& )

a = 1, 2, 3; i = 1, 2 ,..., N, .

P = {Al Ji,A

a,i

Then we have THEOREM 5. An arbitrary orthogonal sum

function

f EX

may be uniquely represented

as the

f =fd + c u%? a where

fd = Pd f E s+& and f(i) a

= U’*‘*f a

E s a,

a = 0, {IX, i}; OL= 1, 2, 3; i = 1,2 ,..., N, .

The representation

is true for an arbitrary

bounded smooth function

Theorem 5 may also be formulated THEOREM 6.

The wave operators u*u

= f,

uu*

cp(t), t E (- CO, 00).

in terms of the operators U(*), U(*): s’? + S: U(*) satisfy the relations = I-Pp,,

HU = UI&,

(38)

where f(I) is an identity operator in s’@*). The demonstrations

of Theorem 5 and 6 are based on the following

LEMMA 1. Let X be a non-singular point of the integral equations (2), (4), (14) i.e., hew’, WI = wg v (ua wa) v WF . Let S, f ’ be smooth functions with compact support. Then the spectral function E(h) is diferentiable with respect to h, and $ (E(h)S, f’) = fdP F;*‘(P, f) Fh*)(I’, f’) 8(Pz - X) +;

j-dp,&,(~a

,f)F,,i(pti

,f’)

%P, - g,i - 4.

(39)

THREE-BODY

COULOMB

415

SCATTERING

Here,

The demonstration of this lemma may be given by the usual method (see, for example, [l]): First, we have to use the expression of the spectral density in terms of the resolvent R(z)

g @(4Lf’) = & ‘$ ((R(Xf k) - R(X- k),f,f’). Then we use Green’s theorem to convert the right-hand integral

X

s SR

R(X, A”‘, h + i0) &

(40)

side of (40) into a surface

R(X”, X’, h - i0) dS, ,

where as usual, the symbol f($/aR) g means the antisymmetrized

derivative

and the integration is performed over the sphere I X i .= R. Taking into account the asymptotic representation (18”), we may represent the integrand as the sum of products QF)Qp) of distorted waves. The terms containing products Q~*‘Q~*’ with differents indices, a # b, vanish in the limit R -+ cc while the diagonal terms Q&*)Qr) give a non-trivial contribution when R --z co. This yields the equality (39). To demonstrate the orthogonality of the wave operators p*n

u(+) h _--- 6oh[ ,I

(41)

we have to use Green’s theorem together with the Schriidinger equation. Let f(P,), f’(Pb) be smooth functions with compact support and equal to zero when E, = Z, , Z, E w’, E, = P2 if a = 0 and E, = pa2 - E,, i if CL= (a, i}. Consider the integral lab = (U,J U,j’). We may represent it as the limit

416

S. P. MERKURIEV

The integrand may be transformed

s

into the surface integral

dX U,(X, PJ U&X, P;) = (Pa” - f’;’ - iO)-’

vR

1

X

u&x,

pa) &

u&x,

pd) ds, .

(42)

The asymptotics of the integrals over the angle variables pa and p8 will be given in Lemma 2 below. Integrating the resulting formulas (43) (43’) over 1P, 1 and 1PL 1, and using the well-known relations m exp {iXtj f(t) dt = 0 , t --ii0 = 27$(O),

h-+-CO,

s-rn

x --f co,

yields the equalities &b = L(Lf’), which prove the orthogonality of the wave operators (41). For the discussion of the integrals over the angle variables put &+‘(X, E) = j.dp U,“‘(X, P)&(P),

Here, f,(p) and fW,i(jJ jc E S(2). LEMMA

2.

/ P 1 = E1J2,E qkw’,

are smooth functions of the unit vectors P and $a , p E P5)~

Thefollowing asymptotic formulas are true if I X I -

03

I(*) = fi(25~)-~” 1P j-3’2 QF’(X, P)f,( +;Q) + IF:‘@‘, P), 0

Q;” = Q,,

&

Ql-’

o =

Q’+’

o .

(43’)

The functions r$*)(&) belong to the class ~~*)(~)(~~*i(~)). The terms IF) and will be described in Section 4. The proof of Lemma 2 is given in the Appendix.

417

THREE-BODY COULOMB SCATTERING

B. Non-Stationary Wave Operators The wave operators described above are called stationary. Non-stationary Coulomb wave operators were defined by Dollard [4] (seealso [ 121)as follows 0,:’ = 4!1 p

‘0

exp {iHtl exp :-iH(n)t

= ?;QII; exp {iHtj exp r -iH,t

A iy,.,(t)l

P,,, .

(44) (34’1

+ iTo(t

Here, Hta) = HI; + V, , and the symbol P,,i stands for the or;hogonal projector onto the subspaceof functions of the form $~~,,(x,)f( JJ. The Coulomb phaseoperators rp,Jt) and To(t) are given by

in the momentum representation. Let IW,i and I,, be the identification operators from P,Y,ifl to X7.; and from .&’ to Z. (seealso [ 11). THEOREM

7.

The non-stationary watic operators are related to the stationary otze~

as,folloivs [p0 = @‘I 0

@’ _ fyq/ .p .,,I n.? n., . 2.i

0,

N= 1,2,3,

(45)

i = 1, 2,..., N, .

It should be noted that the relations (45). together with Theorem 6, implies the completenessof Dollard’s wave operators. The proof of this theorem is analogous to the one for neutral particles, the main modifications being in the following LEMMA 3. (1) Let f (P), f (P’) be smooth,functions with compact .szzpportand equal to Zero if / P I2 = A,. , A,; E 0’. Let

If(f 3f’) = (exp {--iHt)

lJ’*v 0 1

exp (--iH,t

t iTo(

f’).

Then (46) (2) Let fa,#)

andf L,JP’) be the functions

418

S. P.

MERKURIEV

where &.ick> is the Fourier transform of the eigenjiinction &&!l,) and the smooth functions fa,<(pa) h ave compact support and are equal to zero ifpa - E,,~ = A,, Xk E WI. Let P’?&

,L:J

= (exp l--iHt)

UJ:%;.i,

exp { -iH’“‘t

+ iFa,i(t)} f$).

Then ,$pm ~t(ll~itc?.i ,r&:i) = (Pu,&,i

(47)

>P%Zfk.i).

In the proof, we shall concentrate on (46); the proof of (47) is similar. Taking into account the equality exp (iHt}

UA*) = iJA*) exp {iH,t},

which follows from Theorem 5, we get Zj+‘(Jff’)

= (Up)exp

(--i&t}f,

exp (-z&t

+ ivO(t)}f’).

(48)

Consider the integral Zo’f’(t, R, E)

= (27~)-~ SdP dP’f(P)f’(P’) X

s vR

dX Ue)(X,

P) exp (--i(P’, X) + it(F2 - P2) + iF,(P, t) -

In terms of Zi*‘(t, R, l ) we may express limtF+

Zj*)(f,f’)

= Zz)(f,f’)

as the limit

Z2)(f, f’) = tl$m l$ km, --t Zp)(t, R, E). Let t > 1 and R = O(t”), u -C +. Then the variable t is the principal parameter which defines the asymptotic behaviour of the integral (49). Applying the stationary phase method, we obtain t’l,mmf;*)(t, R(t), l ) = 0,

R(t) = 0(t”), v < B.

Hence we may consider the integrals (49) where the integrating over X is performed outside the sphere V(R’) of radius R’, R’ > t”, Y > &. Using Lemma 2 we may represent the integral over angles p, P’ as the sum of products of distorted spherical waves. The non-trivial contribution to the limit t-+F co comes from the six-dimensional spherical waves QF’. Integrals containing

419

THREE-BODY COULOMB SCATTERING

such waves will be denoted by 1(*)(X, 1P 1, j P’ ), t). Integrating variable / X I, we obtain the representation

over the radial

%W = ir t 1 Li2/p, ) >< (=

/ p’ 1 A/ + / p 1 + i())-1-i(qO’2’P1

exp :t i* I

!

In 2 ( P j t - i ___~ 4”T 4lYl I (50)

for non-trivial functional dependence on t. The limit, when t ---f $cc, of integrals over p ’ and p’ ( may be calculated with the aid of the relations (see, for example,

WI) exp { +Xxt} ,I!.% (x f Ql-n exp {in sign f In ( t I] =-= -2&6(x) exp { *:ixt} ,"Tc

(x

f

j())l-in

;yl”i*

in) . (51)

exp {in sign t In 1t 1: = 0.

This yields the equality (45).

4. SCATTERING

OPERATOR

A. Distributions In this section we shall describe the construction of the kernels of the Coulomb scattering operator. First we shall introduce some distributions in terms of which we describe such kernels. We shall denote by t- l - in, t E [0, l] a distribution defined by analytic continuation of the integral si t-“f(t) dt, f E Cm[O, 11, from the domain Re z < 1 [29]. Note that the regularization

may be used for this distribution. We shall denote by (t F iO)-l--in, t E [-I, continuation of the integral

s

11, a distribution

1

-l f(t)(t

in E, E 3~ 0 for E - 0.

f ie)-L-in dt

defined by the analytic

420

S. P.

MERKURIEV

Remember that the (2 --f 2) elastic scattering amplitude represented in the form

Aoli,ai(pa , Pj) may be

where 6, and &,,i are smooth bounded functions, 71, = n,/2 1pa j. We shall associate with this amplitude a distribution which will be denoted by the same symbo1 Aaivoii(jar, p:). In the local coordinates t = 1 - cos 19 with cos 6 = (Ia, $L), this distribution becomes t--l-+. The (3 -+ 3) scattering amplitude A,, has more singularities as was discussed in Section 2. This amplitude may be represented as the sum (53)

corresponding to the asymptotic expansion (28), where the term & is a sufficiently smooth function. Tt has the singularities (1 k, 1-l ) P j)lf2 and (1 ki 1-l 1P’ l)lfs if the Coulomb force between the particles of the pair 01is attractive. Other terms have non-integrable singularities generated by the three-body elastic (3 + 3) processes, as well as single-scattering and double scattering processes. These terms will be considered as distributions given by

(54)

where the following notations have been used: q,, = q,,(P)/2 / P 1, A”, , 2:) and x$’ are sufficiently smooth functions. The singular function 1P - P’ j-5-2igo is to be considered as a distribution. In the local coordinates, t = 1 - cos 8, cos 0 = (p, P’), this distribution becomes t-1-in. In the local coordinates, t = 1 - 2, ) X j-l, the distribution (Pa - PL)-3-iaa also becomes t-1-in. The singularity (kzB - kB2 - iO)-l-i%@ is considered to be the distribution (t - io)- l - in. Finally, the distribution f:“) are defined by fk”‘&

, k:) =tXk

, k@ - iexp (2iargB(l

+ ‘*)I

LI

@a,. R,),

where S($, , x,) is the S-function given on the unit sphere in R3. The two-body amplitude fti for Hamiltonian h has the non-integrable singularities due to the Coulomb potential n, 1X, j--l. In the local coordinates, t = 1 - ($ , &.J, the distribution fn

THREE-BODY

COULOMB

421

SCATTERING

becomes r~l--~?l.Note that the distributions f:’ and I p, - p: (--3-i11* (f~‘,f~‘l (kf, - k,” - iO)-l-iaao depend on different kinematic variables.

and

B. kernels qf the Scattering Operator Let S be an integral operator in the Hilbert space 9 given by the kernels

S”.,j(P,

PA) = -i7?“(2

/ P 1-l)“” 6(P’ - p;’ L Efi,j) A”,,jj(P, ph), (551

S,j.O(pil , P’) = 2i7?“8(p*’ - E,., - P”) A,,,,o(j,, S&P, P’) = -in”’

P’),

(2 1P l-1)3!28(P” ~ P’“) &“(P, P’).

It should be pointed out that the singular kernels A,i.,i and A,,, are distributions defined above. Consider the operator in 9 s = (-J-“(J/“‘, which we shall call the scattering operator. If /E @. f = {J;, ,f,, j], then the vectorvalued function f^’ = Sf^is given by .fn’ = c &J.fb ,

f’ =

f,f’l,

a = 0, {x,ij.

b

THEOREM 8. S is a unitary operator which commuteswith the asymptotic Hamiltorrian fi(, . It is an integral operator whosekernels are given by (55).

The proof of the first part of this theorem is completely analogous to the one for three neutral particles (see [l]). We do not give the corresponding arguments here. The expressionsfor the matrix kernels s,, = UC-)* ,, (/‘-’ h may be derived from the following assertion, which will be proven in the Appendix. LEMMA 4. Letf,,(P, , Pi) be the amplitudesof the spherical wave terms corresponding to the terms f,,,$ and fF) in (42). Then

L.i = E + f,,i ,

E = ps2 - Ei<.j.

ho

= 4W5’2 I P’ I-’ j-dp’ S&(po , P’) f,(P),

.L

= -iCW”‘2/

P 13’2Ei,:‘2 s&C &,,dp, P~J&.&X),

.f& = -i(2n)1”21 P 1-3’2IdP’ S&P, P’) f,(p).

422

S. P. MERKURIEV

For the proof of Theorem 8, we have to represent the integral UJ-)* Ub(+; as a limit of surface integrals on the spheres (R). The limit R + co may be performed with the aid of Lemma 4. It should be pointed out that in contrast to the case of neutral particles the matrix elements of the Coulomb scattering operator do not contain a-function terms. This was also shown by Vesselova who used the existence of the Dollard wave operators [12]. In the presence of long-range forces, the a-function terms are replaced by new distributions. For example, the &functions S(P - P’) and 6(P, - P,‘) are replaced by 1p - p’ l-5-2& and 1p, _ pi pin =, respectively, and instead of the double scattering poles (kia - ki2 - iO)-r we have the distorted poles (ki, - kp - iO)-l-in. It should be noted that the physical interpretation of the matrix elements Sabis the same as for neutral particles. The effective cross section for the process (b + a) is equal to C,, [ Aab 12,a, b = 0, {01, i], where the coefficients C,, are well known (see, for example, [3, 301). The non-integrable singularities of the elastic scattering amplitudes make the complete cross section infinite.

CONCLUSIONS

We have described the principal objects of the quantum scattering theory. Knowing the mathematical aspects of the Coulomb problem, we may investigate numerical methods, from which the physically interesting quantities may be calculated. In particular, the method based on the differential formulation seems to be effective and is mathematically correct both below and above the threshold. For example, the elastic and the breakup scattering amplitudes for p-d scattering may be calculated in this way. The Vesselova integral equations are compact below the threshold. The physically reasonabIe model of p-d scattering was investigated using AGS integral equations. In these approaches, however, there are unsolved mathematical problems above the threshold. APPENDIX

In this appendix, we want to give the proofs of Lemmas 2 and 4. First we shall investigate the asymptotic form of the typical integrals I, = [dj

uy12exp {iE*‘2($, x)} 9 (iv + $- , r, + p, iE1’2 I x I u) g,(j).

(A.l)

Here, p, XGR”, u = 1 - (6, 9, r, = (n - 1)/2, and the integration is performed over the unit sphere P-l) in the n-dimensional space Rn. The symbol d$ means a surface element on P-l), the variables u and p are real, the function gz( $) is smooth and equal to zero if u 3 1, i.e., gt($) = 0

if

($,9)

GO.

THREE-BODY LEMMA

A.I.

COULOMB

423

SCATTERING

For E112 j x j -+ co, we hnue the following

asymptotic

representation

where

The amplitude A(&, p) is dejined as the integral p zz EWj,

4% P) = j-d3 B,(% p) g4$),

(A.3)

where the singular kernel B, is given by B, = / 4 _ j [-2%+2fv.

(A.4)

In local coordinates u = 1 - (j, 2) on the hypersphere W-l), distribution ~-l-~*, u E [0, I].

this kernel becomes the

Proof. On the surface S(+l), let us introduce the spherical coordinate system with respect to the direction R

jQ E p-2),

j = cos 0 @ sin efi, Then the surface element on ,P-l)

cos e - (a, 4).

is

dj = sin”-2 0 dd A d&f = (24’4

du A di@,

where dii? is the surface element on the hypersphere S(n-2). Let us decompose the integrand corresponding to the integral Z, into the sum

First we consider the integral P generated by the term g;(j) - g;(i), and divide the integration domain S(n-1) into two parts s2, and az, where Q; is a neighbourhood of the point $ = 1 given by the conditions i.e.,

mes Sz? = O(/ x I-r+,+V’), L& = (8: j $ - 4 1 < j x j-w+“}, sz,

u

a,

=

vr > 0, v > 0,

(A.3

p-1)

The corresponding integrals will be denoted by Iil) and f$ll. The difference dg( j) = g;( $) - g$(&) satisfies the estimate

I 4mI

d cu,

u = 1 - (a, a>.

(A.61

424

S.

P.

MERKURIEV

Taking into account the estimates (A.5) and (A.6) we obtain: / Zp’ 1 < C mes S, Ty

dg($) = O(l x I- ‘n+“-‘),

O
ez

64.7)

Hence to the leading order, the integral It (I) does not give a non-trivial contribution. In considering the integral 1’) we use the familiar asymptotic form of the confluent hypergeometric function,

da, c, t) - F(c) P(c + F(c) P(u)

- a) exp{--a In t](l + 0(t-l)) exp{t + (a - c) In t}(l + 0(t-l)).

(A4

The second term of (A.8) generates the “spherical wave” asymptotics (A.2). The corresponding amplitude A,(‘) is given by (A.3), where the function d g(j) is to be substituted for the function g*( 3). Then the integral 64.9)

converges. The asymptotic form of the integral corresponding to the first term of (A.8) may be established via integration by parts over the variable U. Only the point u = 1 gives a non-trivial contribution, which we denote by S&). (We do not give the explicit formula for S1il).) Next, we investigate the asymptotics of the integral 1i2) corresponding to the term g9(a). In spherical coordinates, this integral take the form Zp’ = g(i) exp {zP’~ j x I} a,-, ID1du ~~n-l+“(2

- 24)‘*-l

x exp {--iE1/z~ I x I} sp(iv + 42, r, + p, iEli Here g(9) = g*(i), and Q-, f be the integral given by

I x I u).

is the volume of the unit hypersphere S’n-2) [33]. Let

f = g(i) exp {iE1’2 ( x I> 52,-227”-1~1, , where I,, = j-’ du U%--lfw12 (1 - u)-‘-~” exp {-iE1’2 / x I u} v(iv + p/2, r, + p, iE112uI x I). 0

The function (1 - ~)-l-~~ is considered to be the distribution integral 14” can be represented as the sum

f-1--in, t = (1 - u). The

THREE-BODY

COULOMB

SCATTERING

425

First consider the difference dli’). The integrand &i2) satisfies the estimate (A.6) in the neighbourhood of the point u = 0. Then the asymptotics of &j2r may be obtained using the representation (A.8) as in the case of the integral Ii”, The second term in (A.8) generates the “distorted spherical wave” (A.2). Its amplitude is given by Eq. (A.3), in which the function g*( $) is to be replaced by the expression ((2 - L[)rcl

- 2r)l-y 1 - 1,)-l

f”) g(S).

(A.lO)

Only the point u =:- 1 gives a non-trivial contribution P) for the secondterm of (A.8), but we do not give an explicit formula for &P2). To calculate the integral II1 , we may use the equality

.I I*’0 PUP(l - t)-l-iv .+“9+/2 = X%-“&L,

+ iv, q -t I” *- I, At) c/f z-(--iv) &y/2 f 1)

q t p + 1, X) -

n1 + P + q/2) ’

(A.1 I)

which follows from relation (7.623.3) of Ref. [33]. We conclude that the asymptotics of f are given by the sum of two terms. The first term has the form (A.2), where the amplitude A is given by Eq. (A.3) in which the function g;(j) is to be replaced by the expression 2’n-I( 1 - L,)-‘-fv g( 2’).

(A. 12)

The secondterm Gfcorresponds to the contribution of the point t = 1. The sum of the three terms Sf!‘) x 1 U(2l and 8f generated by the boundary point u = I may be proved to vanish. Collecting the sum of all the other terms, we obtain the expressions(A.2) and (A.3). (Such terms correspond to the functions defined by (A.9), (A.lO) and (A.l2).) Now we may give a proof of Lemmas 2 and 4. We investigate the integral I, only. The integrals I,.! may be studied similarly. It is evident that the contribution of the ~(8) classterms in the asymptotics (22) may be represented in the form (A.2)-(A.4). The asymptotics of the terms corresponding to the eikonals 2, Z, and Z,, may be estimated using the stationary phase method [34]. We have two critical points ( j, -f) -m=i.. I for the eikona1.Z. The eikonal 2, (Z,,) has one critical point only which satisfiesZ,, == X / (Z,, = / X :). The critical point (P, 2) = - 1 generates the first term in the representation (43). The other critical points lie in the singular directions, where the terms x0 iJ, and UN, are described by meansof special functions. Hence to prove Lemmas 2 and 4, we have to investigate the integrals over the small neighbourhoods p,), TTYand FIB of the singular directions. In the singular directions p,, and V, , the asymptotic form of the functions x0 and U,, is described in terms of the confluent hypergeometric functions ~(a, c, E). LJsing

426

S. P. MERKURIEV

the explicit expressions for x0 and U, given in Refs. [I 7,221 and remembering Lemma A.1, we obtain the terms in (A.2) corresponding to the distributions As) and A:). To investigate the asymptotic form of the integral la0 generated by the function U,, , we may follow the method developed in the proof of Lemma A.l. All we have to do is to replace (A.ll) by the relation (7.623,4) of Ref. [33]. Then we obtain the contribution corresponding to the distribution A$). ACKNOWLEDGMENTS I am indebted to Professors R. Schrader, R. Seiler and D. Uhlenbrock Institut fur Theoretische Physik of the FUB in Berlin (West). I wish help in preparing the manuscript.

for their hospitality at the to thank Dr. M. Forger for

REFERENCES 1. L. D. FADDEEV, Tr. Mat. Inst. Akad. Nauk SSSR 69 (1963). 2. W. M. KLOET AND J. A. TSON, Nucl. Phys. A 210 (1973), 380. 3. S. P. MERKURIEV, C. GIGNOUX, AND A. LAVERNE, Ann. Phys. (N.Y.) 99 (1976), 30. 4, J. DOLLAIW, 3. Math. Phys. S (1964), 729. 5. A. M. VESSELOVA, Teor. Mat. Fiz. 13 (1972), 368. 6. C. CHANDLER AND A. G. GIBSON, J. Marh. Phys. 15 (1974), 1366. 7. J. ZORBAS, J. Math. Phys. 17 (1976), 498. 8. G. CATTAPAN AND V. VANZANI, Nuovo Cimento Lett. 20 (1977) 465. 9. GY. BENCZE ET AL., Nuovo Cimento Lett. 20 (1977), 248. 10. J. V. NOBLE, Phys. Rev. 161 (1967), 945. 11. A. M. VESSELOVA, Teor. Mat. Fiz. 3 (1970), 326. 12. A. M. VE~~ELOVA, Teor. Mat. Fiz. 35 (1978), 180. 13. E. 0. ALT ET AL., Phys. Rev. Lett, 37 (1976), 1537. 14. E. 0. ALT ET AL., Phys. Rev. C 17 (1978), 1987. 15. E. 0. ALT ET AL., Nucl. Phys. B 2 (1967), 167. 16. V. G. GORSHKOV, Sov. Phys. JETP 40 (1961), 1481. 17. S. P. MERKURIEV, Teor. Mat. Fiz. 32 (1977), 187. 18. S. P. MERKURIEV, Yad. Fiz. 24 (1976), 289. 19. S. P. MERKURIEV, Zap. Naych. Sem. LOMI Leningrad 77 (1978), 148. 20. S. P. MERKURIEV, Dokl. Akad. Nauk SSSR 241 (1978), 68. 21. S. P. MERKURIEV, Teor. Mat. Fiz. 38 (1979), 201. 22. S. P. MERKURIEV, Lett. Mad Pkys. 3 (1979), 141. 23. A. Y. POVZNER, Dokl. SSSR Akad. Nauk 104 (1955), 360. 24. R. G. NEWTON, “Scattering Theory of Waves and Particles,” McGraw-Hill, New York, 1966. 25. L. HOSTLER, J. Math. Phys. 11 (1970), 2966. 26. L. ROSENBERG, Phys. Rev. D 8 (1973), 1833. 27. V. A. FOCK, “Problems of the Diffraction,” Moscow, 1970. 28. R. K. PETERKOP, Sov. Phys. JETP 43 (1962), 616. 29. I. M. GELFAND AND G. E. SHILOV, “Distributions,” Moscow, 1969. 30. R. G. NEWTON, Ann. Phys. (N.Y.) 74 (1972), 324. 31. H. BATEMAN, A. ERD~LYI, “High Transcendental Functions,” Vol. 1, McGraw-Hill, 1953. 32. S. P. MERKUR~EV, Teor. Mat. Fiz. 8 (1971), 235. 33. I. S. GRADSHTEYN AND I. M. RYZHIK, “Tables of Integrals, Series and Products,” Academic Press. New York, 1965. 34. M. V. FEDORIJK, “Sadle Point Method,” Nauka, Moscow, 1977.