Spurious solutions to N-particle scattering equations

2.L : 6.B

Nuclear Physics A301 (1978)

1-14 ; © North-Holland Publishing Co ., Amsterdam

Not to be reproduced by photoprint or microfilm without written permiWon from the publisher

SPURIOUS SOLUTIONS TO N-PARTICLE SCATTERING EQUATIONS COLSTON CHANDLER Physikalisches Institut der Universitdt Bonn, D-5300 Bonn 1, Endenieher Allee 11-13, FRG and Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131 USA t Received 11 January 1978 Abstract : The problem of spurious solutions to the integral equations of N-particle quantum scattering theory is treated . In particular, the Federbush discussion of the Weinberg-van Winter equation is enlarged . It is shown within this framework that the Narodetzkii-Yakubovskii equations, the Bencze-Redish equations, and the Kouri-Levin-Tobocman equationscan all have spurious solutions. Certain model independent results are also presented.

1. Introduction Over the last decade or so a variety of systems oflinear integral equations have been developed to deal with the problem of N-particle quantum scattering systems 1). Uniqueness of solution is naturally an important question for each system of equations, but it is also a question about which surprisingly little is known. It is clear that all of the systems exhibit nonuniqueness at energies corresponding to eigenvalues of the N-particle Schrödinger equation. At other energies, where the extra solutions are called spurious solutions, results are meager 2-10). The aim of this paper is to contribute to this meager literature in two ways . The first is by enlarging on the consideration of Federbush a~ who discussed the threeparticle Weinberg-van Winter equation 11 .12). The second is by presenting some model independent results for the channel coupling class ofequations 9. l0). The model system of Federbush is first. generalized in sect. 2 to systems ofmore than three particles. Within theframework ofthis generalized model rather general systems oflinear equations with connected kernels are then considered . A sufficient condition for the existence of spurious solutions is derived. In particular it is shown how the solution of a relatively simple algebraic problem yields good approximations to the energies at which spurious solutions occur. This technique is applied in sect. 3 to the Narodetzkii-Yakubovskii equations and to the Bencze-Redish equations t4- 13). It is thus demonstrated for these equations that spurious solutions can occur, contrary to a previous claim''). f Permanent address. May 1978

C. CHANDLER

Sect. 4 is concerned with the study ofchannel coupling equations, which appear to be quite an inclusive class of equations" °, 16) . For such equations the approximate energies at which spurious solutions occur can be found by solution of an ordinary matrix eigenvalue problem. As an example, the possibility of spurious solutions to the Kouri-Levin-Tobocman equations 17 .1e) is demonstrated, also contrary to a previous claim 6 .21) . In sect. 5 certain model independent aspects of the problem for the channel coupling class are discussed. Finally, it is to be emphasized that the possibility ofspurious solutions should not, in itself, be considered as a decisive factor limiting the usefulness ofthe equations. The various theoretical uses 22) so far made of the Weinberg-van Winter equations attest to this, as well as the reasonable numerical results that have been calculated on the basis of some of the other equations 1 .23) . 2. The generalized Federbush model In 1966 Federbush 4). devised a model three-particle system for which the Weinbergvan Winter equation has a spurious solution. One of the three particles is infinitely massive, and hence fixed, while the others interact with the fixed particle, but not with each other. The interaction potentials are chosen to be separable potentials, sharply peaked in momentum space. This sharply peaked characteristic essentially fees the kinetic energy of the particles, and hence allows the original complicated operator problem to be accurately approximated by a much simpler algebraic problem. In this way the proof of existence of spurious solutions to the three-particle Weinbergvan Winter equation is essentially reduced to the solution of a quadratic equation for the approximate energies at which the spurious solutions occur. As a generalization ofthis approach, then, consider a system of N particles . Particle N is assumed to be infinitely massive, and hence stationary . The other particles have masses mj, 1 5 j 5 N-1, and momenta pj. The light particles do not interact with each other, but dointeract with the heavy one via potentials ofthe form (2.1) V,(pj) = A,Pj = ,k,1O;'(p,)XO;'(p)IHere the -1j are real numbers, and the Oj(R) are square integrable approximations to the delta function . That is, there is some kj such that

f

lim dpllp(a v)I2 f(p) = fikj), n- m

(2 .2)

for all continuous functions f. Because the operators P1 act on different independent variables, they commute, [Pi, Pj] = 0. (2.3) This commutation property is one of the keys to the working ofthe model.

SPURIOUS SOLUTIONS

3

Channels in this model are specified by which particles are interacting with the heavy particle and which are free. Channel Hamiltonians, as well as the full N-particle Hamiltonian, can therefore be defined in the following way. Let the light particles be indexed by the set A =- 11, . . ., N -1} and let a be a subset of A. Then the channel Hamiltonian Ha, corresponding to the channel in which the particles interacting with the heavy particle are indexed by a, is defined by Ha = HO + E Vj,

(2.4)

The operator Ho is the kinetic energy operator, which in momentum space is multiplication by the function Ho = E (2 mj) -1 IP~j2, I

(2.5)

where units have been adopted in which fii = 1 . The full Hamiltonian with a = A is denoted by HN, and the free Hamiltonian Ho is obtained when a is the empty set. All of the Hamiltonians Ha are linear self-adjoint operators with domains dense in the Hilbert space of square integrable functions. Their spectra are all contained in the half-line

Thus, in particular, the resolvent operators G.(z) =- (z-H )-1

(2.7)

all exist and are analytic in the cut plane z ¢ 9. The first important estimate in this model is a bound on the commutator [Ga, Pj] =- G.Pj-PjGa, = G.(H.Pj- Pf H)G.,

(2.8) (2.9)

where it is not assumed that j belongs to a. Because of the commutator property (23~ the operator Ha in the braces in (2.9) can be replaced by Ho. Then, the specific form of Ho and Pj can be inserted into (2.9) to obtain ' 2 )GQ (2.10) [Ga, Pj] = (2mj)-1 Ga(IPj l P,-Pj lpJI It is now a simple matter to compute the bound 2 (2.11) II[G., Pj]II 5 2xo/d (z~ where go

=

LX

i

lfdpltpj(n)(p)12(2MJ)-2(lpl2_lk 1I2)2~

(2.12)

C. CHANDLER

and where d(z) denotes the distance of z from ®. It is useful that the bound in (2.11) is uniform in a and j. The second important estimate is a bound on differences of the form N-1

where

N-1

d o(z) --- ( fI PI)Go(z)- G,(z) rl PJ, J-1 J=1 G(a ; z) = {z - Eo Eo --_

The property (2.3) implies that

1

_ZJ} -1,

E (2m)-1 Ik11 2. 1

(2.13)

(2.14) (2.15)

da = Ga ( fJPJ
(2.16)

Ilda l 15 (N-1)eo/d2(z).

(2.17)

J

and hence that Again the estimate is uniform in a. Eq. (2.2) implies that so -+ 0 as n -+ oo. Hence, the right sides of (2.11) and (2.17) can be made arbitrarily small. Moreover, it is clear that this bound can be made small uniformly in z for z in any compact subset ofthe cut plane z 0 ®. This uniform smallness is essential to the working of the model. A bounded analytic operator that by taking n sufficiently large can be made arbitrarily small in norm, uniformly in z in the sensejust described, will be called a small operator. Thus [G,, PJ] - small,

(2.18a)

d, = small,

(2.18b)

for all a and j. Clearly the sum of a finité number of small operators is small, and the product of a small operator and an operator uniformly bounded in z is small. Consider now operators of the various forms D1 (z) = P,,G.,(z)P,,G,=(z) . . .P, .G,,,,(z)P,.i,,

(2.19a)

D2 (z) = G,.(z)D1(z),

(2.19b)

D3(z) - D1(z)G, ,+l(z~

(2.19c)

D4(z) = G,o(z)D1(z)G.,,,(z).

(2.19d)

It is assumed that the operators D,, 1 S j 5 4, are connected in the sense that each one of the operators Pß,1 5 1 5 N-1, appears at least once as a fäctor. The property

SPURIOUS SOLUTIONS

5

(2.18), and the algebraic properties

of small operators, allow the factors Pi in (2.19) to be grouped together to the left, with only a small remainder. The same properties can be further invoked to rewrite the Dj in the following form: N-1

D~z) = D~z) rl

Pi +small .

(2.20)

The number D~z) is found from Dj by replacing all Ga by Ga and all Pj by 1. We now come to the main problem of spurious solutions to systems of operator equations of the form Tip = Zip + E KtjT1r (2.21) i where the indices vary over some finite range. In obvious matrix notation this can be rewritten T = Z+KT.

(2.22)

The matrix elements Zit and Ktj are assumed to be linear combinations of the Dj defined in (2.19 with Z1j being possibly unconnected. All of the systems mentioned in the introduction are of this form when iterated sufficiently many times. Spurious solutions occur at z 0 ® for which the homogeneous equation has a nontrivial solution . There may also be spurious solutions for z c- ®, but for such z the operators G.(z) may not be bounded, and the machinery described here for finding them collapses. According to the assumed form of the matrix elements Kid, and according to (2.20), eq. (2.23) can be rewritten X = RSX+ LX.

(2.24)

The matrix elements Rte of R are obtained from Kid by replacing all Dj by Dr The matrix S is defined by Sip

N-1

8iß 11 P1. !=1

(2.25)

The operator matrix L is small. It followsthat 1- L can be inverted, with the result X = (1 +M)RSX,

(2.26)

where M is again small. The necessary and sufficient condition for a nontrivial solution to (2.26), and hence to (2.23), is that g(z) = det {1-R-IGR} = 0,

(2.27)

6

C . CHANDLER

where

= , , MAI «I . . ON'--,IMij1Y'I . . ON'-1) . Suppose now that zo o e is a solution of

(2.28)

g .(z) - det 11- K} = 0.

(2.29)

The assumed form of K insures that this zero is isolated and of finite multiplicity . Let -9 be a neighborhood of zo such that its closure does not intersect 9 and does not contain any other zero ofg.. For sufficiently large n the matrix elements IGU are analytic in 9 and are bounded uniformly in zwith (arbitrarily) small bound. It follows that 9 and n can be so chosen that (g - gjg. is in magnitude less than 1 on the boundary of 9. Rouch6's theorem 19) then implies thatg and g m have the same number ofzeros in 9, each zero being counted according to multiplicity . We conclude, then, that ifg W(z) has a zero zo 0 9, then for sufficiently large but finite n there are spurious solutions to (2.22) at some nearby z. More precisely, the following theorem has been proved : Theorem 1 . Suppose that zo o 9 is a zero of gjz). Then, there exists a finite n such that (223) has a nontrivial solution at some nearby z, say z 1 . Further, the larger the n chosen, the smaller is the difference between z l and zo. The proof that spurious solutions to (2.22) exist is thus effectively reduced to the much simpler problem of fording the zeros of the rational function g,,, . 3. Narodetzkii-Yakubovskii and Bencze-Redish equations The N-particle homogeneous Narodetzkii-Yakubovskii (NY) equations 13 ) can be written in the form Xa2 = ~ i J(bN-1 r- a2)Má3(z-Ho)-1 }Xaz, (3.1) az and the corresponding Bencze-Redish (BR) equations 14, 1 s) in the form a2

b2

e3

N- 1 c-

a2

b2

b2 *

(3 .2)

The symbol $(bN - 1 c a2) is zero if the clustering bN - 1 of the particles into N-1 clusters can be obtained by breaking apart the clustering a2, which consists of two clusters . It is otherwise unity. The remainder of the elaborate notation is that of Bencze 14) and is adopted here without further explanation. The first important, model independent, point to notice is the following : Theorem 2 Eq. (3.1) as a nontrivial solution if and only if (3.2) does . In other words, the NY equations have spurious solutions if and only if the BR equations do. To prove the theorem suppose first that (3.1) has a nontrivial solution 'k°' and de-

SPURIOUS SOLUTIONS

fine

Fa 2 -_

Note that (3 .1) implies that

e2

~(b N _ 1

c a2)1192,

7

(3.3)

,Za2 = M.'=(z-Ho)-1ha , (3.4) 2 so that the 7 = cannot all be zero . That the tl form a solution to (3.2) is seen by using (3.4) to eliminate the ~sz in (3.3). To prove the converse suppose that Fas is a nontrivial solution to (3 .2) and define $°2 by (3.4). Since (3.2) then implies (3.3), not all the X°2 can be zero. Use of (3.3) to eliminate Fal in (3.4) yields the result that the Ta2 form a solution to (3.1). To prove, within the framework of the Federbush model, that there are spurious solutions to both sets ofequations, it is therefore sufficient to consider only the homogeneous BR equations (3 .2). For such Federbush systems the operators Oß2 == Mbz(z-Ho) - 1 are of the form D3 in (2.19c), with the difference that the operators V appear in place ofthe Pi. Another important difference is that they are not connected. In this model the clusterings b2 of particles into two groups are all of the form that one particle is free, the others all interacting with the stationary particle . What is lacking to connect Op' is a factor P1, where i is the free particle in b2. In (3.2), however, there fortunately appears in the sum the restriction provided by 002 actually included in the sum N(bN-1 c a2). This condition insures that in all ofthe the first factor is precisely the potential V, where j is the free particle in a2. It follows that if Yal is a solution to (3.2), then PjYa. = Yaz, where j is the free particle in a2. The homogeneous equations (3.2) are therefore unchanged if the operators Opt are replaced by OP'PP where i is the free particle in b2. In this form the kernel is connected and the considerations of the preceding section apply. In summary, both the NY and the BR equations have spurious solutions at the same (complex) energies, providing they have spurious solutions at all. Moreover, the kernel of the BR equations can be considered connected for the purpose of the Federbush model. Proof that spurious solutions exist is thus reduced to an algebraic problem. As a concrete example of how all of this machinery works, consider the four-particle problem in which particle four is the heavy one. The two cluster channels are then the ones in which particle j, 1 5 j 5 3, is free. With this free particle index used to label the channels, equations (3.2) have the form (2.23) with kernel K' given by 0 V1G,V3G13 V1G,V2G12 KBR = V2G2V3G23 0 (3.5) V2G2V1G12 . In (3.5) the notation

V3G3V2G23

V3G3 v1G13

Gt = (z-Ho-V)-1,

(3.6a)

G ij = (z-H 0 - V-V)- 1,

(3.6b)

8

C. CHANDLER

has been used. It is clear that row i of K' can be multiplied by P, from the left without changing anything. As a consequence any solution Y,1 5 i 5 3, to the homogeneous BR equations must satisfy Y = P, Y,. This property, as mentioned before in the general exposition, has the further consequence that each element in column j of K" can be multiplied from the right by Pj without changing the equations in any way. This renders the kernel connected and allows application of the considerations of sect. 2. The function g.(z) then is the determinant of 1-IZ1, where K-sR

A 2 A3 V 2rr 23 Z 2 Z3r7 3 V23 The functions G, and G,j are given by -

~1~3~1~13 0

A14VA3

Z 1~2 V1 v 12

A14ry2r12

G, = (z- EO -A,)-1,

Gij =

(3.7)

(3.8a) (3 .8b)

in accord with the general rule that the barred quantities are obtained from the unbarred by replacing Vj by Aj and Ho by Eo. To simplify further analysis it is now assumed that all the Ajare equal, with common value A. All of the off-diagonal elements of KI are then equal, with common value given by v = ,12(Z-Eo-A)-1(z-Eo-2A)-1 . The function g .(z) is then easily evaluated to be with zeros zo given by

(3.9)

g.(z) = (1-2vx1+v)2,

(3.10)

zo = Eo +U(3± ,~-3)

(3.1 la)

= E0 +3A

(3.1 lb)

= EO,

(3.11c)

The values of z given by (á.11a) lie outside of 9 and hence correspond to places where spurious solutions occur. It is interesting that the zeros (3.11a), which are of multiplicity two, correspond to nontrivial solutions of the equation 3 Y,

(3.12) Yj = 0. j-1 That such solutions, if they occur, would in general imply non-uniqueness was emphasized by Sandhas 1) not long ago. Bencze 20 ) has further pointed out that the possibility of such solutions invalidates previous proofs e . l0 ) that spurious solutions

not long transition b) for matrix matrix aclass zero HN occur state and is example, final in the left of result correspond as coupling this the independent is ofspurious the and is (4 The of 17 mba (3also they comer given point of the elements matrix The paper W, structure interparticle That 18) operators Tandy lc) form BR equations becomes full the fairly the satisfy and zero by have it class, and such equations Hamiltonian The isato elements KLT (3 solutions the of '0) to insensitive given of matrix of the asolutions (2 with has operator from T, connected be the have BR the b) channel bound that potentials, form noted by the is= choice NY zeros It correspond equations of of with channels The isa(HN-Hb(z-Hiv)-1(z-H pointed (3 embraces isin general W simple not and which KLT)aa to and state matrix are permuting that also which chancel have ones elsewhere after the Wna of an the not BR is = isthe Eo plausible out = ain of artifact zero, particular just Z+WGT = also This matrix Ha the = also all to G ato equations only (HN SOLUTIONS both existence or all' the bbaG6 HN-Hb in coupling has finite are of channels the property above array assumed lies simple, of acases class the conceivable, The particles the ofthe matrix very Hb)a)e choice the form from number invalues Yjvarious isof the indices themselves of exact so 9, are class Kouri-Levin-Tobocman interesting balso given Federbush equations, spurious (3 elements to far principal and The equal The ofthe have lies form of channel exhibited by but a) range clearly nature iterations in eigenvectors that the A1 actually of solutions paper 9, model, which diagonal over the This Hamiltonians, corresponds existence ofZ property but are functions that suggests does is all but they occur, linear within not and there two-body for isthat not of call deeply impor(KLT) iscomin both such to ¢('), that this obthe is the are the the aa

9

SPURIOUS do present The bound viously (3 .11 As model so solutions existence rooted

.

.

.11

.

.11c)

.

.1 .2).

. .11

. .

4. Bencze large equations channel

"

. T

.

(4 .1)

The Tb. where the tant

(4.2)

.), .

. Gba

.

(4 .3)

Finally,

The binations kernel For

.

E a

.

.1) (W

where lower channels .

(4.4)

., .

(4.5)

10

C . CHANDLER

where m, is the number of clusters in channel a, and where V6 is the sum of pair potentials that are contained in H a but not in H6. The indices range over all possible channels. Application of the considerations of sect. 2 is straightforward. Eq. (4.1) is iterated until it is connected. The corresponding homogeneous equation then has the form (2.23) with kernel K' =

(WG)r,

(4.7)

where r is the number of iterations required to achieve connectedness. The function g. then becomes g.(z) - det (1-K) = det {1-(WGY},

(4.8)

= 11 det {Ej-WG}.

(4.9)

r

j= l

The W and G are defined as usual by replacing in W and G the operator Ho by Eo and the operators Vj by Aj. The matrix Ej is proportional to the identity,with the constant ofproportionality being one of the rth roots of 1 . Eq. (4.9) can be rewritten in the form r

g.(z)

= Ij {det Ej~} {det (19 -1 -E; 1W)} . j=1

(4.10)

The zeros of g.(z) are then found by solving the eigenvalue problems equivalent to (4.11) det (5 -1 -Ei 1 W) = 0. As a concrete example ofthe channel coupling scheme consider again the four-body problem as formulated in a KLT channel permuting array. The matrix WG has, in the Federbush model, the form WKLTG =

0 0 V3G2

V,G2 0 0 V2 G3 . 0 0

(4 .l2)

To achieve connectedness three iterations are needed (r = 3). All three of the factors in (4.9) turn out to be equal, however, reflecting a general property of the channel permuting array. Again setting all .lj equal to .1, one obtains g .(z) =

{1-Ä3(z-Eo- .Z)-3}3.

(4 .13)

All zeros of g . are readily computed to be those given by (3.1la) and (3.1lb). Thus, the KLT equations may also have spurious solutions, contrary to a previous claim 6.21).

SPURIOUS SOLUTIONS

II

5. Some model independent considerations There are some model independent statements that can be made about the channel coupling class 9 .'6) . The homogeneous equation ofthe channel coupling scheme is (5.1)

X = WGX. The operator G is defined by (4.3), or alternatively by G = (z-H)-',

(5 .2)

Hb, = Sb,H..

(5 .3)

where If z does not lie in the continuous spectrum of HN, and hence is in the resolvent set of H, one can define Y = GX. (5 .4) Then (5.1) is equivalent to (z-H-W)Y = 0.

(5 .5)

To study (5.5) it is convenient to define the matrix P, (5.6) Pe, = m where m is the number of channels under consideration. Then P satisfies PZ = 1 and is, in fact, a projection . Now the consequence of(4.4) that is important here is that (H+ W)P = HNP = PHN,

(5 .7)

where HN is defined by From this it follows that

HN6c - 6ba

(5 .8)

N'

H+W = Q (H+W)Q+HNP+P(H+W)Q,

(5 .9)

where Q = 1-P. Eq. (5.5) is then seen to have a nontrivial solution if and only ifthere is a nontrivial solution to (z - HN)PY + P(H + W)QY (z-Q(H+W)Q)QY

= 0,

= 0.

(5.10a) (5.10b)

From (5.10) we immediately see that the following theorem is true : Theorem 3.Assume that z is not in the continuous spectrum of HN. Then (5.1) has a nontrivial solution if and only if z is an eigenvalue of HN or of N = Q(H+W)Q.

(5.11)

12

C. CHANDLER

This result was first proved in ref. 9). The more elegant proof presented here incorporates suggestions of L'Huillier ls). Theorem 3 could conceivably be extended to include z lying in the continuous spectrum of HN . The obstacle to doing this is that G may not be bounded, and there mayconsequently be problems associated withthe domainsofdefinition ofthevarious operators. Were such an extension true, the mysterious zero (3.11c) could be interpreted as corresponding to a spurious solution of the BR equations at an energy in the continuum. This follows from the equivalence of the BR equations to a member of the channel coupling class, plus the fact that the zero cannot be an eigenvalue of HN since it does not appear for the KLT equations. It is interesting to see how the Faddeev equations Z), which are the same as the three-particle BR equation, escape the threat ofspurious solutions implied by theorem 3. The Faddeev choice of W is given by .XH.-HO), (WF)na = (1-Sb

(5.12)

where the indices range over the various two-body channels of the three-body problem. Brute force calculation then shows that the matrix elements of N are given by F (N )b. = HoQna .

(5.13)

But Hp has only a continuous spectrum, and hence NF has no eigenvalues. Theorem 3 then implies the absence of spurious solutions. Eqs:(5.11) and (5.12) can also be used in the N-particle problem. One soon discovers that (4.5) is violated unless the only channels considered are (N-1}body channels . Unfortunately, this is of interest only if N = 3. Theorem 3 can also be used to ask what kinds of channel coupling schemes do not have spurious solutions. That is, a particular form for N is assumed that is known to have no eigenvalues outside of the continuous spectrum of HN . Eqs. (4.5) and (5.13) can be inverted to obtain the matrix elements of W .Ha-N  +Q.H, . .-Qb Wb. = W +Nb

(5.14)

QGT = - mQGPG -1 ,

(5.15)

The ambiguity W. in the solution is removed by recalling that the coupled channel equations are supposed to become connected upon iteration. This requires at a minimum that W = 0. Thus, one can assume an N with the correct properties, and see what the corresponding W is. Unfortunately, this approach has not so far succeeded in generating an interesting W. Indeed, in a sense the desired properties of the channel coupling scheme are responsible for the spurious solutions. An important constraint on the operator T has not been obviously built into the channel coupling system . This constraint is which is directly verified from (4.2). If (4.1) and (5.15) are considered together as a

SPURIOUS SOLUTIONS

13

system, then the corresponding homogeneous system is given by (5.1) and QGX = 0.

(5.16)

Combining (5.16) with (5.4) yields equation QY = 0 as a supplement to (5.10). This clearly leads to the following theorem : Theorem 4. Assume that z is not in the continuous spectrum ofHr . Then, the homogeneous system (5.1) and (5 .16) have a solution if and only if z is an eigenvalue of HN. Another way of looking at this theorem is to observe from (5.7) that z-H-W maps the subspace defined by the equation QY = 0 into itself. On that subspace z-H-W is always invertible for z in the resolvent set of HN and hence no spurious solutions occur. From this point of view the Faddeev equations are an anomaly. Because of an inspired choice of W, the operator N has no eigenvalues and hence (5.15) is indirectly enforced . It would of course, be extremely interesting to know other useful choices of W with the same property. As a final remark it is well to emphasize again the point made in the introduction, that the presence of spurious solutions is an inelegant, but by no means fatal, aspect of any set of equations. It is a pleasure to acknowledge the hospitality and the financial and technical support provided during the course of this work. The various institutions involved are : the University of New Mexico ; the Central Research Institute . for Physics (Budapest) of the Hungarian Academy of Sciences ; the Physiakalisches Institut der Universität Bonn ; the Fulbright program ; and the Minna-James-Heinemann-Stiftung, in collaboration with the NATO Senior Scientists Programme. The author is particularly indebted to Gy. Bencze and W. Sandhas for many interesting remarks on the subject of this paper, and to M. L'Huillier and A. G. Gibson for interesting correspondence. Note added in proof. I am indebted to Prof. V. Vanzani for pointing out that his work is erroneously evaluated in the introduction . Vanzani 7) has extended the work of Newton s) to obtain for the various homogeneous equations considered here the nontrivial result that the kernels factorize. This establishes the general possibility of spurious solutions. The bulk of the work presented here should, therefore, be interpreted as a demonstration that the possibility can be actually realized. See also .the recent preprint : K. L. Kowalski, Green function equations, factorization properties and spurious solutions in channel-coupled multiparticle scattering theories, Case Western Reserve preprint (1978) .

References

1) W. Sandhas, in Few body dynamics, ed . A . N . Mitra et a!. (North-Holland, Amsterdam, 1976) 2) L . D. Faddeev,'Mathematical aspects of the three-body problem in quantum scattering theory (Israel Program for Scientific Translation, Jerusalem, 1965) 3) O . A . Yakubovskii, Sov . J . Nucl . Phys . 5 (1967) 937

14 4) 5) 6) 7)

C. CHANDLER

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