Recent developments in few-particle scattering theory

465~

Nuclear Physics A416(1984~4650480c North-Holland,

Amsterdam

RECENT DEVELOPMEN'TS IN Fl?W-PARTICLE SCATTERING

THEORY

K.L. KOWALSKI Department 44106 USA

of Physics,

Case Western

Work since 1980 on dynamically

Few-b&

methods

description

we have a lot to say, provide the particular imposed.

painfully

truncation

Nonetheless,

they are viable

complete

are distinguished

tation of a complete

Reserve University,

scattering

a self-consistent

of the physics

conventional

theories

from I_-__ conventional

of the physics.

Ohio

in reviewed.

ones by their represen-

The latter ones, about which calculational

characteristic

methods

Cleveland,

scheme only after

of the method has been

have been of interest

lately because

and people -do calculate with them (two attributes

that have been

absent from few-body efforts

tion of the low-order integral equations

approximations

for N > 4) and because

the clean extrac-

they suggest from full sets of scattering

has proven to he fairly difficult.

How good are the standard algorithms

and what should we look for in attempt-

ing to improve upon them? A hint can be found in the following observation': 1,. ..When the standard methods are applied to rest reactions they can give acceptable result% only because of the flexibility introduced by the use of effective interactions. T"nese difficulties should be borne in mind when evaluating the apparent success of nuclear reaction theories." Can we define and then systematize actions? tempting

Most of this review is dedicated to answer this question

and the notation

ignores the formulation

to truncated

multichannel

2. THE OPTICAL Although it means

scattering

theory.

Ref. 3 can serve as an introduction

used here.

of Chandler

plicit in the C-G approach

of these effective

The otherwise and Gibson

are valuable

inter-

to work over the last three years at-

based on N-particle

last reviews are given at Eugene', concepts

the calculation

(C-G)

The to the

excellent review in Ref. 4 5-7 . Many of the ideas im-

in sorting out the ambiguities

endemic

problems.

POTENTIAL

the optical

quite different

potential

(OP) is the archtypical

things to different

there has been an extraordinary

people.

effective

By whatever

surge of interest of late in the OP.

portion of this new work is a completion

and modernization

formalized by Feshbach' for distinguishable particles. "This work was supported in part by the National Science

interaction,

criterion, A major

of the type of OP Foundation.

466c

K.L. Kowalski

A two-body gffective the elastic

scattering

interaction

Particle Stuttering

from an arbitrary

antihermitian

part is supposed

channel due to inelastic

the amplitude

for

objects: fl(;',$")T(&;) E-E"+iE

-I- (d 2') i

An OP differs

Theory

(ET), UC;',:) generates

of two composite

T(;',;) = U&j

EI by its singularity

(2.1)

* and reality

structure; its

to reflect the loss of flux from the elastic

processes.

A .I vopt(~ ) = Vopt(z)

Disc ici 1 opt Elastic upon

/Few

This is equival.ent to the stipulations ,

(h.a.)

,

(2.2a)

7

(2.2b)

cut = 0

the OP operator

V where z is a complex energy, which ensures that op?f' vopt(z) is h ermitian below the inelastic threshold. While it is easy to find

EI's satisfying

(2.2b) by separating

unitarity

cut', the hermitian

difficult

to satisfy.

The Feshbach'

formalism

T o @)

TR a’“’

= Vopt(B,z) -I- vopt(~,z)

I

Vopt(6,~)

= I+? + “’

(2.4) above the threshold

to question

by Tobocman IO.1

tic unitarity

(2.2a) for Vopt (B,z) which

G@

proposed

QB Va

,

reactions

(2.4) of the re-

has been called in-

technique

and the hermiticity hennitian

( 2.3)

[The validity

removes the elasof V ' impLies

for z not in the spectrum

can be identified with resonant questions

for the

(energy averaging,

of

The threshold. 8 structure .

e.g.), the major

(2.4) is its calculation and a systematic way of doing this was 9,ll . We emphasize that the definition of (2.4) is en-

only recently

tirely independent presents

9

we take to be below the inelastic

of Voptw

where

,

limit.

for rearrangement

is therefore

Apart from interpretational problem with

PD T8 B(z)

Q8 [z - QB H Q&l

from Voptw

QB H Q, that, for simplicity, pole singularities

particles,

The projection-operator

cut structure

is more

,

where z is taken to be E + i0 in the scattering nowned

(2.2a)

of (2.1) - (2.2) for the scat-

(6) of distinguishable

= VB -I-5" G(z) VB

f

across the elastic

(h.a.) requirement

is a neat realization

tering of two bound clusters elastic operator,

out the discontinuity

analyticity

of any formalism

a formidableobstacle

Feshbach's

1962 paper* includes

obtain an effective

operator,

identity

(total) Hamiltonian,

For the generalization transition

used to calculate

to the generalization

of (2.3),(2.4)

it.

Particle

identity

of (2.3) and (2.4).

in the projection

operator

formalism 12c,12d

but not an optical potential one requires

a (anti-) symmetrized

T(g), formed from a class (G) of exchange-equivalent

to

K. L. Kowalski 1 Few Purticle Scattering

rearrangement

operators,

Theory

467C

TB,i .13c; (2.5)

where A

is an exchange operator corresponding to the permutation that maps 838 A the canonical partition, B, into B E 6. Note that T(E) depends on the off-shell form of TBTB as well as

An antisymmetrized

T&z) Only matrix

B , although the scattering amplitudes depend on neither.

OP operator,

elements

= U&z)

can be defined in terms of T(a^,z) as

+ U&z)

G@

T(:,z)

.

13

(2.6)

of

VoptG’e)

are required

* U(B,z),

z PB U(i,z)

in the elastic scattering

tions of TB B(Z) still yield different

P8

(2.7)

problem but different OP'S~~'~~.

off-shell

The "symmetrical"

exten-

AGS14 off-

shell extention, G=fi

b,aGa+G

yields an OP satisfying elastic

scattering

for the so-called satisfying

AGS b Tb,a Ga

(2.2) and so that below the inelastic

is described

by a hermitian

post or prior extensions,

neither

(2.8)

’

(2.2a) or (2.2b), because

potential.

threshold

the

This does not work

which yield effective

interactions

the Tb(t;(z) are not h.a., while the

Tra(z) arelza. AGS is not unique in this respect and any nonpathological, 12 (1 symmetrical", T will generate an OP with the desired properties ; Bencze b,a and Chandler' have generalized the work of Ref. 9 using such a Tb a. The connetted-kernel

(CK) OP formalism

proposed

Ref. 10 using a C-G-type'coupling

scheme is a trivial application of the work of Refs. 6,9. The OP's introduced 15 by Adhikari are incorrectly defined in terms of the inversion of an operator that cannot possess

an inverse.

In view of the proposals (2.2b).

If we introduce

transition

operators,

of Ref. 16, we show how easy it is to satisfy only

the channel

coupling scheme Va = 1 vapc, then the post

satisfy'

T(+) = i- ; s c-1 + I- ; P T(+) where 2 = (Ga6 a,b), S,,, state projector,

= 1 , P = (P,6a,b 6,,,), Pi is a two-cluster(~~~~d

and (Q = I - P) T=V+VGQT=V+I’~QV.

We assume nonidentical particles for simplicity. T(+) not (2.2a). c1 c1is half-shell equivalent to F(+) = r a,a

(2.10)

Clearly r satisfies

(2.2b) but

(2.11)

K.L. Kowalski

468~

/ Few Particle

Scattering

Theory

so r is a non-h.a., formalism-dependent (on the choice of V) EI. With the a, CY prior extension we obtain similar results but with r replaced by Tt (given by

(2.10) with V -f Vt,t z transpose). 9 case .

It is trivial to extend this 'co the

antisymmetrized

The antisymmetrized of two-cluster

forms of I't appropriate

to a given Pauli-equivalent

set

partitions are the "new class of OP's" proposed by Adhikari, 16 , but according to our requirements for an OP the I',Tt are not

Kozack and Levin OP's but EI's. Feshbach

Even in the nonidentical

case Ta u is quite distinct

LJ, c1and depends upon the choice of vale.

rise to ?g&

that violate

is not enough. -_

unitarity

The preceding

discussion

Note that the same approximate depending

upon whether

The general properties

T,Tt may give

threshold:

(2.2b)

also shows that the particular

Ref. 16 is really irrelevant

function formalism usedin t ed status to r,i- .

amplitudes

Approximate

even below the inelastic

from the

and provides no preferr-

r, e.g., will yield different

on-shell

OP formalism

(2.11) or the standard9

of r and implicitly

wave

elastic

is used.

Tt (cf. Ref. 17) are studied

in Ref.

9.

18 have shown that the curious enFinally, we note that Bencze and Chandler 19 ergy-independent OP 1s the abstract form of Feshbach's OP appropriate for time-dependent

scattering

tity and unitarity The properties 10.12,20-22,

theory and thus carries no new physical

questions

still remain unanswered

of t‘$(i,z)

the Feshbach effective

h ave been investigated

and CK, Refs. 9,13,23,

cludes all effects of particle resonance

Hamiltonian

techniques.

identity,

structure,

content. Iden-

in this formalism. using conventional,

This OP satisfies

is formalism

(2.2), in-

independent,

incorporates

and is related to the antisymmetrized

by a nondynamical

Refs.

Feshbach

transformation.

These last elastic

properties, along with some applications to multiple 12~ and in22 have been developed using the conventional formal operator scattering

algebra that would involve illegitimate

operator

inversions

if the scattering

limit (S.L.) z + E + i 0 were taken prematurely.

For example,

"solution"

Ta ,(z) = V'+

[l - Vn G,(z)]-' V'of

does not exist in the,S.L.,

the LS equation

for complex z it can be Learranged

although

the

'JaGa to

Ta a(z)

yield the

"closed form", TC1 c1(z) = Vcc + V' G(z) Vcl, which 11s well defined on the cut. ternatively,

this heuristic

procedure

can be discussed

Al-

in terms of LS nonunique-

ness as merely an algorithm for projecting out all but the desired particular 11 10 has questioned this and has insisted, in effect, but solution . Tobocman without

substantiation,

that all intermediate

fined.

The assessments

of various

are based upon irrelevant

criteria.

inversions

must also be S.L.-de-

OP's in Ref. 10 appear incorrect Unitarity necessarily

one must be careful; e.g. the treatment

requires

in Ref. 24 is not generally

in that they the S.L. and correct.

K. L. Kowalski

All

1 Few Particle Scattering

of the recent work12c~20~22~25

ing (MS) formalisms

fail to achieve

all of the intermediate

combining this while

scatterings.

contains no restriction

cluster states.

46%

identity with multiple taking identity

scatter-

into account

in

The Green's function

cluster channel of interest that appears Refs. 12c,20,22

Theory

G for the twoR in all of the constituent operators

on the permutation

Thus, any approximation

in

symmetry of the S-

scenario necessarily

involves uncer-

tain assumptions

about the role of these unsymmetrical states. Surprisingly, 26 the well-known KMT formalism has the same defect e'ren in the case where pro12c,20,22,26 in jectile identity is ignored. The tricks that have been proposed the KXT case in order to retain a MS structure ing the full Hilbert states.

space and bringing

The cumulant-type

AGS-or-post-generated failing to identify

expansions

tors requires

OP's with identity

target

for either the

also have the serious shortcoming

the role of the heavy-particle

an unphysical

symmetrized

used in Refs. 12c,20,22

the case in Ref. 25, but here the avoidance

exchange

pay the steep price of re-enter-

back unproperly

exchange

terms.

of antisymmetrized

split of the two-nucleon

subsystem

potentials

of

This is not opera-

into direct and

parts.

The virtues of the post off-shell extension (as contrasted to 12c,20,22,27 AGS) put forth in Refs. are based on circular reasoning concerning the generation

of the "correct"

unsymmetrized-target-state N-particle

formalism

particle

low-order

ed wave function

approximation

is deceptive

ciations with the elastic 3. EFFECTIVE

because

transition

the two methods

operator

channel

Ultimately tinguished

{a$,...

rearrangement

,..., B

where either

i

transition aB=

by ambiguities of recovering

operators

uaB(A)+

channels

asso-

the widely

used

in the definition some version

of

specifies

are then determined

7 Q(A) YF.A

UNg(A)

referring

characterized

to a dis-

by the pro-

ranges span the relevant model space.

G P T YYYB

Ua3(A) is g' xv en and (3.1) determines

tension for PuTTaBPB

concerning

defines a set of ET's, U uB(A), of two-cluster

the union of whose

T

refer to different

theory.

the CRC method

set A =

jectors P,,P

(CRC) method motivated

and the related difficulty

the CRC from an N-body

to the

(see Sec. 3).

INTERACTIONS FOR REARRANGEMENT 15b,28 have recently appeared

coupled-reaction

of the

to yAGS yields an equation for the projectopt close to that of the RGM method. This

A spate of articles

of the approximation

because

to some of the MS approximations 23 identity .

that is tantalizing

"close encounter"

this is ambiguous

It appears one really needs a full-blown

to give credence

OP with fullaccountof A natural

MS structure;

problem.

.

from

’ T

UB

The

(3.1) or a given off-shell

ex-

47oc

K.L. Kowalski

It is easy to construct fined (3.1).

For example,

/ Few Particle Scatredrrg

Theory

(for specious

reasons) u a3 (A) that lead to ill-de-

the hermitian

choice

IRef. 28d, Eq. (6.1311

implies

1 - Uwrong E = s G-1 E, so Eqs. (3.1) cannot be inverted. (In Ref. 15 -1 it is implicitly assumed that S exists.) This is why the “recovery” in Ref. 28d of the wave function is incorrect.

form of the CRC within

The reason294

the U aB

for a formalism

function

is being -forced to yield approximate

l $‘2

of

designed

the wave function

for the projections,

correct formalism

is incorrect is that PB/$ (+) > of the full wave

wave equations

for the components,

in the model space H7:. This strategy

tory and does cause difficulties primary

an otherwise

that this recovery attempt

is contradic-

in Ref. 288 but is really irrelevant

to the

thrust of Ref. 28d.

More promising

choices for U can be obtained in a less -~ ad hoc manner. ct$ jA) way is to attempt the consistent solution of the Schrgdinger 2% (wave) equation in fflr , but this is plagued by the nonorthogonality (N-O) and

The traditional

possibly,

overcompleteness

(O-C) difficulties

that have sparked much of the re-

cent controversy. Dealing with this is an important aspect of the C-G 5-7,28k theory . The device of the Moore-Penrose (M.P.) generalized inverse 29 has also proven to be effective in the two-cluster CRC case 28P*q* A such that 1 (A) PB/$a> = 0, so that the -1 PB are not independent. It is in such cases that the M.P. inverse, &f , of the (A) (A) bounded operator 1 PB , e.g., M-l (1 P4) = PT, where Pn is tbe projector O-C occurs when there are components

is introduced to deal with the N-O contributions. ll' A major question is whether O-C appears for physically realistic situations. 30 The arguments of Cotanch and Vincent strongly suggest that it does not; the

onto H

counterexample posed in Ref. 28p violates translational h B, it has been shown12b that no O-C occurs.

invariance.

When A =

Pauli-class

Birse and Redish28pyq

ing to the projections pectation

clarify

several CRC-related

issues by providing

a

-1 I$a(+) > of the components and a reciprocal mappIQa> = Pg Ad

unique definition

lb,> = Ps/$")>.

Most of their work is based on the ex-

that O-C is or "almost is" a problem.

Their effective

interaction

is

found to be

= pa UBR oa where V

c1

= Pr (H-Ho)

(3.3) is non-h.a., hermitianis directproof

1'"Pi M-l P

Pr, and H is the Pm-space

it is free of A-class

expected

(3.3)

B

to generate

effective

elastic unitarity

Hamiltonian.

Although

cuts and because

the correct model space unitarity

H is

relations;

Birse28q

has ex-

a

using (3.1) and (3.3) would be more convincing. h tended the result of Ref. 12d from R to an arbitrary set A by finding an expres-

K.L. Kowalski

defined by (3.1)

sion for

471c

/ Few Particle Scatteri?lg Theory

with AGS for

There has been much interest

T% B13Y28d.

in the comparisdn

of the (low-order)

CRC (RGM)

with approximations

to connected-kernel (C-K) equations or to other approaches. 31 tests indicate that CRC (RGM) calculations for a three-body

Some numerical

system are superior

to undistorted

C-K calculations

space, but seem only to imply that a specific lar formalism

on the same L2 function

technique

designed

for a particu-

(CRC) does not work equally well with other formalisms.

the CRC (RGM) is regarded and/or justification This embedding

useful enough to investigate

by embedding

its possible

Overall

improvement

it within a full scattering theory. 28k. m what is certainly a super-CRC

has been accomplished

formalism,

viz. the C-G method'; this approach is compared with other methods 28m to salvage some of Ref. 15b results in a general in Ref. 28~. An attempt CRC formalism

that bears some similarity

to the C-G method.

The approach of Ref. 28d with a h.a. EI free of the model-space cuts is complete [The reduction proposes

in the sense of providing

procedure

clustered

a set of C-K equations

of Ref. 28g reproduces

sets of successive

unitarity for the EI.

some results of Ref. 28d and

effective

interaction

equations.

Cf.

Ref. 28c.l

Although it is clear what approximation to yields the low-order e; CRC, it is not at all evident how to recover this from the C-K formalism. The

study of systematic approximations to the C-K equations for has only %S 23 begun ; note that although approximate solutions will be free of A-space unitarity

cuts, the h.a. property

may not be preserved.

Birse and Redish 28~ show how the lowest-order version of the CRC is a distorted tions in the wave function covered

the speculation

potentials,

a good approximation

proximation

to the undistorted

ing potential

is dynamically

a true embedding

equations,

The physical

C-K equations.

much, but not all, of

circumstances

Recent

C-K APPLICATIONS

under which this

nor is the equivalent

ap-

The freedom of choice of distort-

ad hoc and so the demonstration --

of Ref. 28~ is not

these results may lead to C-K equation-

numerical

BSA is a poor representation

The formulation

irrelevant

to the CRC and at the very least indicate

of the naive BSA to the CRC.

4. FURTHER

the famous BSA) via a suitable

are not specified,

proof; nevertheless,

based improvements

undistorted

(basically

thus rendering

about CRC vs. C-K.

constitutes

edfrom

version of the so-called precursor BRS equa32 by Levin . The low-order CRC is re-

form advocated

in the pole approximation

choice of distorting

(H 2 H) to their

approximation

TO REACTION

the inequivalence

work seems to suggest 33

of the physics

that the

.

THEORY

of the CRC (RGM) seems simplest

in terms of wave function

and insight has been achieved using wave function formalisms obtain34 16,28p,32 involving . Levin3' has exploited the equations

C-K theories

472~

KL.

Kowalski

/ Few Particle

Scattering

Theory

the components3

I +CY(b)>= (Gt2-l) ba [ p a>

,

(4.1)

32

A

to as "true" in that if where G” = G + ; Vt Gt and ?S = VS, that are referred ta, then only I$, > contributes asymptotically to the o + fi

b = R (two-cluster),

well-labeltransition. This nonunique attribute, is probably better designated ____ ,. 3 ed and requires that the kernel Kt = G Vt become connected after a finite 34 number of iterations . The C-K equations for /i, (b) > are somewhat cumbersome (-1 = E-1 S t Vt (Gt E-l) the relationbecause of the t-operation and because T ship to physical complications VGS

E-l)

transition

encountered

amplitudes

choice of components

This leads to the (e.g., T (+) =

is a bit contorted. The seemingly

in Ref. 32.

(G 6-l)C+p,aI$

more natural

c1> is3 independent

of b (not well-

labeled or "true") and equal to lQa >. Given the operator representation 16 Gt E-1 when V is label transformthe antisymmetrization of formalism is easy ing. Few-cluster

models for reactions

derive their physical

stances where a few (n") clusters of particles s o^ = dominant35 in the explicit EI's.

partition) '

that the possibility

dynamics but may be represented

Ref. 36a (PR) is an ambitious

of few-cluster

models;

relevant mathematical PR formalism

of their breakup implicitly

are discussed

with an explicit MS structure

in circum-

version

consistent

theory

of PR, while some of the

in Refs. 29,36c.

is obtained

(clusters

can beignored

via the intercluster

attempt at a physically

Ref. 36b is a BLKT-type equations

validity

are so tightly bound

of the

A version

in Ref. 36d forarbitrary

n* as well as n" = 3 specializations. The PR (and BRS) approach for "6 = 3 is 0 0 also explored in Ref. 36e. This may be viewed as the formal recovery of -_ ad hoc three-cluster

models.

We do not review the considerable work on the latter. 37 (based on two-particle-connected equations ) approach to few28g,o,35,37 is adopted by Vanzani -et al . The limitations arising

A different cluster models

from indexing by chains of partitions few-cluster

model equations

directly

(COP) have been removed and the leading involve the relevant physical

transition

amplitudes. 5. N-PARTICLE

SCATTERING

Chains of Partitions:

THEORY The scattering

equations

proposed

by Yakubovsky,

eJ

-al3,4 are labeled by COP and involve a complex organization of operators leading to intimidating derivations partly due to the fact that the combinatorics of COP have not been worked into as convenient

forms as that for partitions.

icant step towards understanding

the COP structure

studied

An important

further

compositions

in Refs. 37,39.

A signif-

is taken in Ref. 38 and is

technical

advance involves

the de-

of the partition-labeled operators V a, Ha, Ga, etc, into chain-ele38 which have been placed into convenient forms in Ref. 39 and mentary-components

K.L. Kowalski /Few Particle Scattetirzg Theory

then exploited

to develop chain-elementary

the scattering

in its most finely-decomposed

various

scattering

equations

C-K resolvent

can be derived

39

473c

equations

representing

form and from which all of the . The infamousYakuhovsky 40 recur-

rence relations are circumvented and the inductive development advocated by 41 is justified. Haberzett14' has found partial decoupling of the AGS

Sandhas

equations

so the only transition

cluster partitions into clusters

operators

corresponding

of fixed numbers

contain the suppressed

of particles.

channels.

Partition-Labeled be expressed intuitive

particle

particles,

The structure

of connectivity

while

of sets of partitions.

operators

underlying

a reduc-

results re-

have been obtained

scattering

by

theory can

The former codifies our

the latter is exploited

The classification

by their connectivity

blems that are investigated

particles,

Some combinatoric

in terms of graph and lattice theories.

motions

properties

This is achieved via EI's that

is obtained.

for identical

Theories:

are labeled by those two-

split (n,N-n) of the N particles

In the case of identical

tion to a single integral equation levant to COP equations 43 Karlsson .

appearing

to a definite

to study the

of partition-labeled

leads to nontrivial

in Ref. 44 using the partition

N-

combinatorial

lattice;

pro-

these re-

sults have been applied in quantum field theory and relativistic quantum me45 chanics . 46 A number of papers depend crucially upon the remarkable properties, Refs. 36c,44,46d, various

of the matrix17

submatrices,

to invert the BRS equations Watson-type

multiple

constituent

operators)

the restrictions

equations N(N-1)/2

(ii, b = 1 if agb

equations

obtained

little or no coupling.

equivalent

b) and its

and to derive the

New inversion

(but N-body properties

in Ref. 46d and these are exploited transition

operators

of

to obtain

that possess

In Refs. 46a,b (The derivation

there is not "new".) great emphasis N-l rather than(2 - 1) C-K equations

that are dynamically

equations

with a connected-kernel

first in Ref. 46b.

for the antisymmetrized

here do not contradict

and = 0, if a7

These results are used in Ref. 46~

to obtain the Rosenberg

scattering

of E are proven

new C-K equations relatively

h,,b

square or nonsquare.

of the BRS

is placed upon the attainment for the two-cluster

of

amplitudes

to the BRS equations; the "inframinimal" claims 47 coupling theorem and the realistic case of

the minimal

identical particles causes problems unanticipated in Refs. 46a,b. [The infer48 that the minimal set of two-cluster partitions is also maximal is false.]

ence

The essential tensions46c

simplicity

of Rosenberg's

can be appreciated

C-G Theory:

A number

equations

via graphical

and their multiparticle 4,49 .

ex-

arguments

of important developments and applications of the G-G5 6,7,50 . Elementary derivations of the C-G

theory have taken place since 1980

K.L. Kowalski 1 Few Particle Scattering Theory

414c

(-) p (and variants) for the projected transition operators Tba 3pb Tba 7,51 that depend on the existence of X -1 = (1" Pa )-1, where A iz

equations

have appeared

some set of partitions

(In the C-G theory A = all and X = JJ"; in Ref. 51, A = -1 two-clusters, X f hl.) If this is the case, then since I = X-11* GcPc G c ,(-I = Va + Vb G I Va = G;l G Va, we obtain the C-G equations and ba -1 T = Pb va P + Pb Vb x 1 G P T (5.1) ba a ccca * c -1 50 C-G type equations have been found that do not contain X ; a C-K version of all

the C-G equations plications

has been derived7

with identity.

of the C-G approach

that appears, however,

The real spirit,

is to circumvent

to possess

and the possibly

some com-

tremendous

power,

the C-K problem via the construction

of an

appropriate

sequence of a@roximate transition operators that -do satisfy C-K 50 equations . At this stage C-G seem to be jn the unique position of

integral

having formulated and consistent

a comprehensive

scattering

set of approximations

theory replete with a well-defined

and a solid backdrop of rigorous mathema-

tical theorems. Bugbears

Pathologies:

persist

in few-particle

solutions

are a potential,

equations

with kernels of high connectivity;

spuriosities Chandler equations

are obtained

in Refs. 52.

are characteristic 53

theory.

factorizations

indicating

Some of the spuriosities

of Federbush

Spurious

for reduced

C-K possible

pointed

model rather than the reduced

out by

C-K

.

Bencze and Chandler

54

have shown that the much-abused

(L.I.) holds in the weak topology, studied

scattering

but possibly not serious problem

disproving

Lippmann identity 55 ; the L.I. is

earlier claims

in Ref. 46b, and in Ref. 56 the L.I. is taken as a weak limit and used

to derive the LS equations. A few years ago the very foundation the seeming existence

of standard

C-K theory was threatened

by

of scattering

solutions of the homogeneous C-K integral 57a equations for some special situation . These arguments were shown to imply a 57b contradiction with a known theorem and, more incisively, to involve the violation of the necessary‘ requirement lim K(E) /Y(E)> = K(E + 0) 10

,

(5.2)

E+c where K(E) is the relevant

(singular at E -f 0) kernel and IY'>= lim

leading to K(E + 0) 1'0 # IY>

/Y(E)>,

[It is claimed in Ref. 46b that 57c the results of Ref. 57a result from an improper use of the L.I.] K.S.P.

claim to demonstrate two particles

and no paradox.

that (5.2) is not necessaryforthe

type of problems

(e.g.,

field and, in general, noninteracting subsystems 57b They also claim Sloan's observation actually they consider.

in an external

of N particles)

supports their argument.

K.S.P. propose

scattering

integral

equations

that

K. L. Kowalski

presumably

/ Few Particle Scattering

do not have the difficulties

415c

Theory

that bother

them.

Evidently

we will

We should also hear more about the pro10 limits, that are claimed by Tobocman to exist

hear more about this as time goes on. blems, also involving in manipulations

singular

of more conventional

operator

groupings.

N > 4: Ref. 58 contains a review of the 4-nucleon

scattering

problem and

describes calculations carried out using Hilbert-Schmidt (H.S.) techniques. 59 has proposed an N = 4 version of his N = 3 zero-range scattering theory.

Noyes

Nuch recent N = 3 work concerns 60

separable

representations

of the off-shall

.

input into the N = 4 equations

For the N = 3 system itself, calculations 62 61 Eyre -et technique and the HS method have appeared.

using a new iterative 63 have used a cluster-expansion al _. employing

an elastic

tion to the Watson

channel EI.

formalism

to carry out model calculations

The tests of the single-scattering

and KMT OP's for 7-d scattering

approxima-

in Refs. 64 converged poorly

enough to call the MS series for the n-A OP into question; it would be inter65 2 esting to compare this with the results for a factored tp .L -dlscretization 31 are used with some success in an N = 3 model in Ref. 66 for treattechniques ing continuum

effects

in contrast to the failure of the finite-basis expansions 67 . In more formal work, Kouri '* finds Faddeev-type

for the N = 3 BKLT equation equations

with BKLT effective

adjoint Hamiltonians AGS equations

interactions;

in quasi-Faddeev

with a three-body

the spectral

equations

potential

properties

are investigated

is reconsidered

ofnon-self-

in Ref. 69;the

in Ref. 70 looking

at

the effect of a V3 bound state. A distinctive coordinate partially

three-body

approximation

space has been formulated separable,

potentials

N = 2 t-matrix

formalism

for local potentials

in Ref. 71 employing

in

the method of the

of Ref. 72a that also has been applied to

with absorption 72b (See also Ref. 73).

Separable

approximations

to

N = 2 t-matrices

are studied in Ref. 74a which is a critical study of a pre74b viously proposed technique , while Ref. 74~ consists of the detailed elabora74d tion of a method that seems to possess remarkable convergence properties, exact bound-state/on generalization are proposed

-half-off-shell

to multiparticle

characteristics,

amplitudes.

and the possibility

New iterative

techniques

of

for N = 2

in Refs. 75.

Other Results:

The N-body permutation

symmetries

are confronted with the

single transition calculational proposed

operator V + V G V in Ref. 76a; this leads to a variational 76b strategy . Multiparticle variational principles have also been

in Ref. 77.

The semi-classical expansion of N-body Green's functions 78b to obtain N = 3 spectral sum rules; is developed in Ref. 78a and applied 78c such sun rules were previously found . Refs. 79 contain recent results on

time delay.

In Ref. 80, H is approximated

by a sequence

of self-adjoint

bound-

476c

ed operatots essary

leading to a general scattering

condition

equations

to obtain resolvents

is proposed

in Ref. 81.

theory of approximations.

An interesting

proof of the unitarity

[l,N] Pade approximant

has been given by Balazsx2.

on a line are obtained

in Ref. 83.

scattering nificant

A nec-

such as that in (2.4) from scattering

New results

of the

for scattering

Rigorous characteristics of low-energy 84 are studied by Belle _et -.. al . The BKLT equations which play a sig-

role in the review of "calculable"

methods

Ref. 48, have been applied to reactive scattering

in many-body

scattering

in

in Ref. 85.

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