The orthogonal projection method in scattering theory

ANNALS

OF PHYSICS

111,

330-363 (1978)

The Orthogonal

Projection

V. I. KUKULIN Institute

of

Nuclear

Physics,

Method

in Scattering

Theory

AND V. N. POMERANTSEV A4oscow

State

Unioersity,

Moscow,

USSR

Received August 3, 1976

This work gives a detailed account of the orthogonal projection method in the theory of two- and three-body scattering, which is based on the employment of orthogonal projecting pseudopotentials. The method is applied to a number of physical problems, of which the following are the most important: the improvement of convergence and the rearrangement of Born series to make them convergent at low energies in the presence of bound states in a system, as well as the consideration of the Pauli exclusion principle in the scattering of composite particles and in the integral theory of direct nuclear reactions. The properties of eigenvalues of kernels of the equations obtained are investigated and the conditions for the convergence of their iterations are derived. For the three-body problem, the general case of three different particles is considered, as well as two particular cases, namely, two particles in the field of a heavy core and three identical particles. The proven theorems are illustrated by numerical examples. Other useful applications of the proposed approach are listed and discussed, in particular, those in solid-state physics and in the theory of electromagnetic transitions. The approach suggested is compared with those of the other authors and the prospects of using the developed formalism are discussed.

1. TNTR~DUCTI~N

In many quantum mechanical problems, there arises the necessity of considering the wavefunctions not in the complete Hilbert space *, but in the subspace 8Q C Z’, which is the orthogonal complement of a certain (finite- or infinite-dimensional) subspace &‘r specified constructively using orthoprojector l? A broad class of such problems involves the consideration of Pauli’s exclusion principle in atoms, molecules, nuclei, and solids. Indeed, the main role of Pauli’s principle is to forbid a part of phase space accessible to the system, i.e., to suppress the projections of the total state vector on the “forbidden” subspace. Many problems of this kind can be solved using standard methods, e.g., the Feshbach projection operator technique [2], which enables one to derive the equation for the projection Q# from that for the total vector $. For some problems, however, a direct application of this method does not yield the desired result. For example, this is the case with the manybody scattering problem, when the particles being scattered are themselves the composite ones and interact through certain optical potentials containing two-particle states forbidden by the Pauli principle. Such a problem involves as a particular case the integral description of deuteron-induced direct nuclear reactions on the basis of a three-particle model. In such a problem, it is required not only to project the vector 330 0003-4916/78/1112-0330$05.00/0 Copyright AU rights

0 1978 by Academic Press, Inc. of reproduction in any form reserved.

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331

# on the all.owed subspace 60 but also to eliminate all virtual transitions to the forbidden states, i.e., to construct two-particle t-operators in the subspace be. It is just to solve such a problem for the three-body system that the method of orthogonal projection by means of orthogonalizing pseudopotentials (OPP) was suggested [l]. The method permitted the Faddeev equations (FE) to be modified consistently enough [3], without including many-particle projectors, but renormalizing only two-particle t-matrices to remove the forbidden virtual transitions. Another large class of problems of the similar type arises in variational calculations of excited states in systems with Hermitian Hamiltonians, when the functions for the excited state:s are sought in a subspace orthogonal to the lower-lying state vectors. The OPP :method was applied to a simple variational problem of this kind [l] and it turned ou’t that the method provides a numerically stable, effective computational scheme. In a subsequent work [4], the present authors have shown that the projection method using OPP is a highly convenient and effective tool for rearranging the Born series in the theory of potential scattering; it enables one to obtain a convergent series at low energies and with any number of bound states in the system. The idea of the rearrangement is the same as that used in the well-known Weinberg method of quasi-particles, that is, the separation from the kernel of the Lippmann-Schwinger equation of a finite-rank operator corresponding to large eigenvalues of this kernel. In contrast to Weinberg’s ;approach, however, we have explicitly used the orthogonality of all the scattering functions to the functions of the discrete spectrum of Hermitian Hamiltonian. In the case of changeover to the integral formulation, this is equivalent to the substitution of the kernel (1 - r) K, where r is a finite-diemensional projector, for the kernel K. Tt is obvious that in such a case the maximum (modulo) eigenvalue (EV) of the kernel does not increase, i.e., for such a substitution one should expect, in general, that the convergence range of the iteration series of the integral equation will broaden. Fr40m the standpoint of convergence, the choice of the projector on the invariant subspace of the operator Kin the capacity of r, as has been done by Weinberg (Weinberg’s “ideal choice”), would be optimum. Our scheme is, however, much simpler to realize, yet it enables one to obtain well-convergent Born series. The present work deals mainly with the application of the OPP technique in order to improve the convergence of iterations of equations in the theory of two- and threebody scattering. How the OPP technique is applied to include Pauli’s principle in the composite particle system is briefly described in Section 4. In Section 2 we introduce the orthogonalizing pseudopotentials, derive a modified Lippmann-Schwinger equation (LSE), and, on the basis of estimates of EV of the kernel of this equation, prove the main theorem of convergence of the projected Born series. Whereupon the general theory is illustrated by several numerical examples. We have investjgated the cases of the Hulthen potential and separable potentials as treated generally. Then we also extend the method to the case of “noneigenstate” projection and compare it with the Weinberg method of quasi-particles [5], as well as with approaches of Scheerbaum et al. [6] and Saito [7], which have been developed on the basis of the Feshbach projection technique. 595/IIII2-5

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AND POMERANTSEV

In Section 3 we generalize our formalism to solve the three-body scattering problem and discuss the general case of three different particles and two important particular cases, namely, two particles in the field of an infinitely heavy core and three identical particles. Section 4 presents some applications of our approach. There we describe a modified system of Faddeev equations (FE) for the system of three composite particles, which has been obtained in [3] and which includes Pauli’s principle, and we write explicitly the addition to the kernels of the integral equations due to the composite nature of the particles involved. Also, we discuss an important question of how one can include in the scattering problem the information contained in the structure of the bound states, which may be the main part of the Hamiltonian, by eliminating to a certain extent the ambiguity present in the description of scattering states. We also discuss there a possible application of our approach as a substitute for the OPW method in the theory of band structure in solids. In the Conclusion, we dwell upon some aspects of the practical application of the developed approach and present a comparison of our method with the other methods used in the theory of many-body scattering and also based on the iterative series. Appendix A presents a more vivid proof of the main theorem of Section 2, which allows a number of additional properties of EV’s of the modified kernel of LSE to be found. In Appendix B, we prove one useful inequality for the kernel of the FE’s.

2. TWO-BODY

SCATTERING

2.1. The Orthogonal Projection Method

Consider the SchrGdinger equation (SE) for the scattering problem (E - HI Y?E= 0,

E>O

(1)

with an additional condition of orthogonality (9’ I h> = 0,

(2)

where p) is an arbitrary normalized vector, and the Hamiltonian H is Hermitian and has only negative EV’s of the discrete spectrum. The exact meaning of the problem (l), (2) will become clear later on; for the present, we show that it can conveniently be solved by going over to a modified Hamiltonian (pseudo-Hamiltonian) I? [I]: if=

H-/-XI’,

r=

I TXYl

(3)

and the real constant h should be brought to infinity in the final solution. Indeed the solution $E of the SE with the pseudo-Hamiltonian I? (E - HII h> = X I T)(T I &;i>

(4)

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is equal to

& = 4~+ c(X)G(E)1 v?

(5)

where G(E) = (E - H)-1

is the Green function (GF) of the initial Hamiltonian and the #E solution of Eq. (1). (Naturally, it. is assumed that the asymptotic behavior of s/~ and G(E) is consistent, i.e., G(E) = Jim,_, G(E + in) and #E = #E+). The constant c(h) is equal to

Then the admixture of the state v in the solution $E is

Thus, at h --f 00 c(h) = -(y I #d/(y I G I v>, lim,,, a rigorous projection is achieved, rqR = 0 and

(y / GE) = 0, i.e., at X + to

We note that, along with the solution (4), Eq. (3) has the homogeneous solution z,&= G(E,) 9) at a definite energy E, , which can be determined from the condition h-l = (g’ ] G(E,,)j v). But this solution and the corresponding singularity in the renormalized GF are unessential for us. If the vector g, is the eigenfunction (EF) of the initial Hamiltonian H corresponding to an energy Ef , then for the Hermitian Hamiltonian the condition (2) holds automatically and the relationship $E = #E should hold for the scattering states. Indeed, from the expression (7) there follows the required coincidence of the scattering wavefunctions. In the particular case when there is no interaction potential, i.e., at H = Ho, G = G, , and t,/~= Z/J,,, from formula (7) we can reality obtain the solution &o(E), which was introduced, in a general manner, as the complete orthonormalized basis of the scattering theory in [6a] and, in a particular manner, in [7]. Following the authors of [6], these functions can be conveniently called the orthogonality scattering functions (since they describe the scattering due to the orthogonality condition). The authors of a series of works [6] made extensive use of such orthogonality scattering functions to construct a basis convenient for describing the nucleon scattering by nuclei and for other purposes. In our approach the orthogonality scattering is defined in the standard way-as the usual smttering by the orthogonalizing pseudopotential V,, = iIF,,, , i.e., by a separable potential with an infinite constant.

334

KUKULIN

AND

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The extension to the case of projection on several mutually orthogonal is obvious:

vectors vi

In a way similar to the solution (5) we now obtain

where

(1 - hG)ij = 6ij - h(yi / G(E) / yj).

t8b)

If h ---, co, the constant h is eliminated from the expression for $& and we again have the rigorous orthogonalization p+%
i = 1) 2 )...) N.

(9)

Taking into account (8a) and (8b), the expression (8) may be written as

where the operator [r(l - XC) r) is detied pulled on the vectors vi , and is

solely in the N-dimensional

subspace,

Similar operators will often be used in what follows, without specifying their ranges of definition in each particular case. Approaching the problem less formally, we indicate that the two-particle SE (1) with the orthogonality condition (2) is, as pointed out by Saito [7], a highly useful and simple model in studying the scattering of composite particles (tritons, ol-partitles, etc.) from nuclei with the inclusion of Pauli’s principle (the orthogonality condition model). In such a case, the states vi will be the forbidden states of the system and the scattering wavefunction must be orthogonal to them. In order to construct an adequate scattering theory in such a model with the orthogonal projection, it is necessary to define the corresponding scattering operators, which should satisfy the condition of orthogonality to the eigen- or noneigenstates of the initial Hamiltonian. Consider first the GF e(E) = (E - A)-l for the pseudo-Hamiltonian A and establish its relation to the initial GF G(E) = (E - H)-I. Making use of the standard operator identities for the GF, we have

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whence we immediately obtain (I? = G + GrA(r(IThe admixture nonorthogonal

XG)r)-lrG.

(11)

to r is

rf5 = r(r(i

- hi) r)y

rG=

0,

i.e., at h + co the i: is also orthogonal to the subspace of “occupied” ~5 = G - Gr(rGr)-1rG.

(12)

states (13)

When the functions P)~included in rare the EF’s of H, as in [I], the operators r and G commute, and the renormalized GF (: goes, at X -+ co, to the truncate GF [l] (14) That is, from the spectral expansion of the initial GF, G(E), pole terms, corresponding to the “forbidden” states, are eliminated. In what follows, this will be referred to as the eigenstate! projection. The result (12) remains valid, i.e.,

Now we compare the method of pseudopotentials with that developed by Saito, who has proceeded from the Feshbach projection formalism [2]. Let, as before, r be an N-dimensional orthoprojector. We designate the N-dimensional subspace of values of r as br, and its orthogonal complement as dQ (Q = 1 - r). Then the problem (l), (2) may be formulated as follows: to find the solution of Eq. (I) in the subspace 6’0. Refinements of this formulation lead to various formalisms. Tn the Saito approach, in particular, the problem is formulated as follows: to find eigenvectors (in the generalized sense) of the continuum of an operator generated by the Hamiltonian H in the subspace 80, i.e., the operator

HQQ= QffQ. This leads to the equation proposed by Saito [7] (E-H,,)@=

0.

The GF of Eq. (15), i.e., the resolvent of the operator H,, f~r,,)-~, satisfies the equation

GS = (E -

G"=G-G(rH+Hr-rHr)G"

(1% , which we designate as

336

KUKULIN

AND

POMERANTSEV

and we obtain G” = G - Gr(rGr)-l

rG f r/E = C?+- r/E.

(16)

From (16) it follows that at E # 0 the solution of the Saito equation (15) coincides with the solution (8) for the equation with the pseudopotential if X -+ co. At E = 0 the solutions (15) do not satisfy’ the orthogonality condition. In particular, at E = 0 it is always possible to add any vector g, E br to the sclution (15). This is of no consequence for solving the problem (l), (2). Tt is, however, important that when we use such a projection all forbidden state poles of the GF are not eliminated, as in the pseudopotential approach, but are grouped at E = 0 (see (16)). This prevents us from using the GF of the Saito equation, G”, and the t-matrix associated with it to take account of Pauli’s principle in many-body problems, where the forbidden states must be eliminated from the off-mass-shell two-particle amplitudes. To improve the convergence of Born series at low energies, this method is quite useless, too. If, however, one adds the orthogonality condition r#, = 0 to Eq. (15), as has been done in [6], and all the scattering operators that ensue from Eq. (15) one takes in the orthogonal subspace d O, then the results are obtained which coincide with ours. In terms of the projection formalism, at h -+ co the GF of our approach i: is, as is really seen, simply the inverse of the operator (E - H) in the subspace bQ, i.e.,

~7= Q[Q(E- HI Ql-l Q

(17)

r?Q(E - H) Q = Q.

(18)

or

Thus, in the context of the scattering theory the mathematical trick with the infinite coupling constant h is simply an effective way of inverting the operator in the orthogonal subspace. Such a trick turns out to be convenient also because, having determined (: and the corresponding t-matrix, we can work with the complete space H without worrying about projection any longer. All peculiarities corresponding to the states included in the projector rare automatically eliminated at X + co. It is interesting to note that, since h is strictly excluded from all final expressions (in the limit h -+ CO)and the rigorous projection is carried out just in that limit, such an approach is partly similar to the classical method of rounding singularities in the GF when instead of the real energy E we use its complex value E + ir and go, in the final expressions, to the limit E ---f 0. It is a familiar fact that in such an approach E vanishes from all final expressions and the requisite conditions for outgoing solutions are automatically fulfilled. The introduction of the pseudo-Hamiltonian Ais, however, highly convenient also in variational calculations of atomic and nuclear states [l]. When pseudopotentials are used in this way, X is not removed from the final expression, but is merely thought to be large enough to ensure the necessary projection in the minimization of the relevant functional. Then the orthogonal projection arises from the fact that, at large positive X, those components of the state vector $ that overlap

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THEORY

appreciably with the functions included in the projector will steeply increase the energy of th[e system, and, as a consequence, the weight of those components will continually decrease in the minimization; i.e., the orthogonality will be reached. To conclude this section, we note that the orthogonalizing complement hr (at X + a) wa:s first suggested by Courant [9] within the framework of variational methods to introduce additional orthogonality conditions to the functional extremum problem. Tt was suggested as one of the particular cases of the method of penalty functions-now a well-known method in optimization theory. 2.2.

A Mot@ied Lippmann-Schwinger

Equation and Born Series Rearrangement

Let us apply the orthogonal projection techniques as outlined above to rearrange the Born series in the theory of potential scattering in order to make it convergent, if it is not, or to make the convergence more rapid, if the initial series converges slowly. The main idea of the orthogonal rearrangement of Born series is quite transparent. If a system, which is described by the Hermitian Hamiltonian H = H, + V with the attractive potential V < 0, has a bound state F,, with the energy I& , then, as is known [lo], at small positive energies the Born series diverges, because in the vicinity of the point E = 0 at least one EV of the kernel G,V of the LSE exceeds unity [5, IO]. Making use of the fact that at positive energies the solution of the SE is orthogonal to y,, it turns out possible to rearrange the Born series in such a way that each of its terms will be orthogonal to y, . Such a modified series is the Neumann series for the modified kernel fI?,,V, whose EV’s turn out to be smaller at negative energies than the corresponding EV’s of the kernel G,V. If there are no bound states apart from q0 , it turns out that the orthogonal series converges at any negative energy, up to zero energy, and so it does at least at low positive energies. The rearrangement itself is conveniently achieved by means of the orthogonalizing pseudopotential [ 11. We proceed from the pseudo-Hamiltonian I?=H+XT,

H = H,, + V.

(20)

Then, using the usual identities, we get

where, according to (13) co = ‘,‘y (E - Ho - X)-l /

= G, - G,r(rG,jr)-’

rG0

(22)

and G, = (E - H&l.

(23)

Further, defining, as usual, the wavefunction of the scattering problem by the equation 6 = 1;1i i&(E

+ ic) 9,

(24)

338

KUKULIN

AND

POMERANTSEV

where C$is the incident plane wave, we obtain the LSE for $

+,, = Q. - G,r(I’G,I’-l

FiD,

(26)

is a “plane” wave orthogonalized to pi (to be more exact, this is the solution of a free equation in the orthogonal subspace go). If vi are the EF’s of the Hamiltonian H with Ei < 0, then from (14) and (24) we have

That is, in this case, Eq. (25) gives the scattering wavefunction for the initial Hamiltonian H and is thus a rearranged LSE. Before setting out to determine the modified t-operator, we note that in projecting on the eigenvectors, i.e., on the invariant subspace, it is possible to write an infinite number of expressions for the projected t-matrix, which will be coincident on the mass shell, but which differ in the positions of their singularities. While using such renornormalized operators in many-body problems (see Section 4), we shall obtain considerably differing many-body amplitudes. In this respect, the OPP method leads to a complete eliminate of the contribution from the forbidden states to the many-body amplitudes. In [ I] the modified t-matrix was given by t = -G,’

+ G;lcG,l

(27)

and it was shown there that in the case of the eigenstate projection it coincides on the mass shell with the ordinary t-matrix. This t-matrix satisfies the equation following from Eq. (21) for the GF f = VC&G,l - r(rGr)-1

rGG,l

+ vcq.

(28)

Here we have used the relationship (29) which we shall also need in the following consideration. From (12) it follows that v+g hri:

= -r(rGrp

rG.

In the case of eigenstate projection, the second term in the right-hand

(30) side of (28)

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THEORY

turns into -.PG$ and gives no contribution on the mass shell. Therefore (on the mass shell) we have the following modified on-shell LSE

f = VG,G,’ + Vf5”i.

(31)

Iterations of (31) give, however, asymmetric terms and it appears more convenient to proceed directly from (27) using as G the iterations of (21) provided, of course, they are convergent. Then the series will be symmetric. In particular, the zero-order approximation corresponding to the orthogonality scatterring function $, is f. = -r(rG,r)-l

and the first-order approximation il = V -

(32)

will be

(TGJ-l

+ (rG,r)-l

r

rG,V

-

VG,r(rG,r)-l

rG,VG,r(rG,~)-l.

That is, the -matrix element of 2 in the first order is (33) From these expressions it follows that, unlike the usual Born series, in the case of symmetric iterations (32) the zero-order approximation includes no interaction operator (instead it contains only the projector on the discrete spectrum). For the noneigenstates projection, the modified on-shell t-matrix yields of course another phase (see below). The transition operator corresponding to the pure orthogonal scattering was minutely discussed in [6a] where the contribution of this term to the total scattering was particularly discussed using the rectangular well potential as an example. The term &,was found not to represent the total scattering but, it proved to give an essential portion of the elastic scattering, thereby decreasing the scattering due to the very potential. Tlhus, it is felt that the iterations of the rest should converge much better than those of the initial LSE. Nevertheless, this problem has not been studied yet elsewhere, and it is the resolution of this problem that consitutes the essence of this section. 2.3.

The Convergence of the Rearranged Series

We now demonstrate that the iterations of the rearranged Eq. (25) with the kernel c,,V (or Vc, , see Eq. (31)) will converge more rapidly than those of the initial LSE with the kernel G,V and that the iterative series for the modified kernel e,V may converge in the presence of any number of bound states (if they are included in the projector r).

340

KUKULIN

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The convergence of the rearranged series is determined by the EV’s of the kernel e,Y. If the G,V is a compact (completely continuous) operator (this implies certain restrictions on the potential [lo]), then, obviously c,,V also compact and thus the series converge provided all EV’s of the operator e,,v are less than unity, I Zi, j < 1. From now on it is always assumed that the compactness condition is fulfilled. In [4] the properties of EV’s of the kernel G,P’ have been studied in detail for the attractive potentials V < 0, when r is one dimensional. Here we derive the main properties of in a shorter way and without the above restrictions. In the further consideration E < 0. In such a case, -G, is positively defined ; therefore, the kernel K = G,V can be symmetrized:

K = (-Go)lie K,(-G&1/2; K, = -(-G,,)li2 V(-Go)'/" = KS?.

(34)

The EV’s of the operator K coincide with those of the Hermitian operator KS and, therefore, are real. Now consider the kernel of Eq. (25), R = e,K We write C?,,as co = -(-G,,)l’”

(I - P)(-Go)li2,

(35)

where

P = -(-Go)liz T(TG,F)-1 r(-G&l/z is evidently the orthoprojector.

(36)

Using Eq. (35), we find

k+= -(-Go)1/2 (1 - P)(-GG,)'/2V = (-Go)llz (1 - P) Ks(-Go)-lj2. Thus, the EV’s of the kernel R may be found from the equation

(1 -P)KIX)=aIx>. Acting with the orthoprojector

(37)

P upon both sides of this equation, we get BP] x) = 0.

(38)

The condition (38) means that E has the N-fold zero EV, while the remaining eigenvectors and eigenvalues are such as one of the Hermitian operator

R, =(I - P)K,(l -P) and therefore are real and can be found from the minimax principle. Specifically,

(39)

341

SCATTERING THEORY

Thus, we have that all the EV’s of the operator i?, , and satisfy the inequalities amin

<

gi,,

<

consequently

-Ymas.

those

of

K,

(40)

Moreover, i Zi, / -E--r, 0, as Ij e,vii ‘E---m 0. These results are valid for any orthoprojector lY If, however, r contains all the bound states of the system, and only them (this case will be referred to as total eigenstate projection), the equality d = 1 is impossible. Indeed. &(6-‘) = 1 means that there is a pole in the resolvent (1 - e,,V)-l

and, con-

sequently, in the GF (? = (1 - e,P’pl G, - TG,, as well at the energy i. But this is impossible, because we have assumed that the poles, corresponding to all bound states (and there are no others), are excluded from G. Thus, Ei, # 1 and 101, j + 0 at E + - CCand, therefore, taking into account the continuity in E, we conclude that at E -< 0 6, < 1. Hence it follows that for the rearranged Born series to be convergent it is sufficient that the condition cymin > - 1, or taking into consideration the monotonic character of the functions n,(E), the Nmin(O) > -1 be fulfilled. Thus we have proved the following Providing the complete continuity of the operator G,V in the case of the THEOREM. total eigenstate projection, the iterative series of Eq. (25) with the kernel e,V converge at all energl;es E < 0 if the absolute value of negative EV’s of the operator G,V does not exceed unity, i.e., if the condition a,,i,(O) > - 1 is fulfilled. In particular, for the attractive potentials all 01, > 0 and the rearranged series always converge. By virtue of continuity and at small positive energies E there must be a range of convergence of the rearranged series if ami, > - 1. Moreover, several numerical examples indicate that the convergence retains in the entire range of positive energies (see example (a)). For the present, however, we are unable to give general proof for the case of arbitrary positive energies. A more visual proof of the main theorem is given in Appendix A. Let us now illustrate the properties of the projected kernel e,V by two examples. (a) Conrergence of iterations for the local potential. Let us investigate the modified eigenvalues d,, at E > 0 for the Hulthen potential V(r) = - V&l - exp(-r/a))-‘, where the values of the parameters V,, and a chosen so that (V,,aa . 2~ = g) the Hamiltonian would have one bound state with the energy E, = (a” * 2p)-l[(g - 1)/212. The EV’s a,(E) and the corresponding EF’s 9(K, E) for the Hulthen potential are calculated in the analytical form and are well known [II]. The formulas for the calculation of the corresponding EV’s & of the modified kernel G,P’ are also obtained in a simple form. The trajectories of the first two EV’s a,(E) for the values of the coupling constant g = 2.0 and 3.9 are shown in Fig. 1. The second value corresponds to the case when the second energy level is slightly above the threshold (it appears at g = 4), while the ground sta.te is strongly bound. This case seems to be the most unfavorable (in the

342

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Imcc

FIG. 1. Trajectories of EV’s LX,(E) (dashed lines) and a,(E) (solid lines) corresponding to original, and orthogonalized, Golf, kernels for S-wave Huhen potential with the coupling constants g = 2 and g = 3.9. The numbers at the curves are energies in the units chosen (see the text). G,V,

FIG. 2. Argand plot for S-wave amplitude of scattering by Huhen energies. The numbers denote the order of orthogonal iterations.

potential

(g = 2) at two

presence of one bound state) for the convergence of Born series at small energies, because al(O) is rather large (= 3.9). Nevertheless, as can be seen from Fig. 1, the largest EV of the orthogonalized kernel C?,,Vd,(E) -=c1 and the series for the kernel C?,v converge at all energies in this case, too. We have also calculated the s-wave partial amplitude at several different energies and at the coupling constant g = 2. The obtained values of the scattering amplitude are shown against the number of iterations in the Argand plot (see Fig. 2), from which one can see that the rearranged Born series converges well both at low and high positive energies. (b) Convergence of iterations for the case of the separable potential. illustration, we consider the separable potential v = -K/b

I g>
As a second

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343

Then the t-matrix will be t = V/(1 - 6(z)), where a(z) = -(d&)(g

I G, I g>,

and the Born series is a numerical series in powers of 6(z). If a z = E,, < 0 there exists a bound state, then 6(O) > S(E,,) = 1 and the series is divergent at low positive energies. Thle modified Born series for t‘is obtained by expanding in powers of 8:

8=

I GoI yi)-l KgI GoIg)(y I GoI v,>

-4+4<9)

- l

(41)

where spis the wavefunction for the bound state

Making use of the relationships

= (6 - 1)y/N;

(P’ I GoI d = (Z - Ed1 f ~(1 - s)), (42)

where ‘)’ =

N2[K/2p(Z

-

&,)]-I,

we have

&@--l)r-(6-l)y-ll’ When z = 0, (p’ 1G, I v) < 0 and, in accordance with (42) the denominator in (43) is positive. Consequently, 8(O) < 1, as 6(O) > 1. On the other hand, taking into consideration the Cauchy-Bunyakovsky inequality, it follows from (41) that 8(O) > 0. Thus, / s(O)/ < 1 and the modified series for fin powers of 8 converges in the vicinity of z = 0. We note that (43) may be written in the form (1 - 8)-l = (1 - 8)--l + y, whence it is clear (see the definition of r) that, as a result of renormalization, the pole at z = E,, is eliminated from the t-matrix. Furthermore, even at sufficiently high energies, at which the ordinary series in powers of 6 converges, we obtain, due to the rearrangement, a more rapid convergence of the Born series. 2.4. Noneigenstate Projection Let us now apply the developed formalism to the projection upon functions which are not the eigenfunctions of the Hamiltonian. In so doing, we are solving the ordinary SE without the additional orthogonality conditions, unlike the way it has been done

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in Section 2.1. This will be done in order to improve considerably the convergence of the iterative series in those cases when the system contains no bound states, but has only the near-threshold singularities (virtual states or resonances) or when we just wish to make our series converging more rapidly. We then proceed from the conventional concepts [5, 12, 131, i.e., we assume that if the components, describing (possibly, approximately) the virtual states and resonances, are separated out from the total wavefunction, the remainder of the function will be small and may be found from perturbation theory. We, however, use a different computational procedure. Let the projection operator .P = Cr=, 1 &(qi [ include along with the vectors of the usual bound states, certain vectors yi, which approximately describe the virtual states or the resonances of the system (on the condition that (qpi 1TV> = 6,j). Let Q = 1 - r and, as usual, rQ = Ql’ = 0. We now represent the wavefunction as two mutually orthogonal parts:

Further, following equation for *c

the classical method of Feshbach [2] one can easily derive the

(E - H) $0 = (E - H) r(r(E

- H) r>-1 r(E - H) $0 ,

(44)

and, since our aim is to obtain a solution $a explicitly orthogonal to #r, we, in compliance with the general recipe, introduce the projecting pseudopotential xr in the left-hand side of Eq. (44), and then we obtain the integral equation $

0

= d + GG-ll-G-l$o

_ d +

C?HrH$,

r(E - ~)r'

rvr

where $ is the general solution of Eq. (E - H - AT),,, H - AT)-l IA-a,. Solving the obtained equation for $. , we have

+0=J+

r(E-

eHrH$ HHcH)r

(45)

1,6= 0 and G = (E -

'

(46)

This equation may be derived in another way, namely, by projecting the complete GF: G=QGQ~QGr+rGQ~rGrEGooi-Gor~GrofGrr

(47)

where the meaning of symbols is obvious. Then, after simple manipulations one can readily derive the relations between the projections of the GF in (47) and the modified GF, e: G o. = e + eH9(E)Hc; Gro = -9(E)Hc;

Go,=

-GH~(E);

G-r = =%Q

where B(E)=

r{r(E-

H-

~G~)rr)-v.

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345

If we now Idefine the complete # by the usual equation

we easily derive the splitting of the form

where the expression in braces represents the #o component, while the #r component is represented by the last term. Let us now take into account that the modified function 6 satisfies Eq. (25) whose iterations converge the faster the more components (in general) are included in the orthoprojector I’. It is important that in many cases (see [6, 71) even the “free” solution of z,&,, i.e., the zero-order approximation, describes I$ rather well. At the same time, it appears that in these cases the “free” projected GF, G, , gives rise to a good approximation for G and, consequently, also for #a , i.e., in such cases, the scattering function will be determined from (48) adequately. It would be of interest to explore this scheme for specific examples (in this connection see [13, 141). Comparison of Eqs. (45)-(48), obtained for the projections of the complete wavefunctions, with the analogous formulas found by Shakin et al. [6, 131 indicates their identity provided, of course, that, in their projection formalism, by the modified GF, e, we understand not the operator (E - QHQ)-l, but the operator

Q[Q(E-WQI-IQ = QF- QW-'Q. On the ot.her hand, the noneigenstate projection may turn out useful also in the case when the eigenstate projection alone is insufficient to ensure the convergence of the iterative series of Eq. (25), for example, in the presence of strong repulsion. Then, having included in r appropriate terms assuring the elimination of large negative EV’s one can achieve a good convergence. Therefore, in such an application our method turns out to be rather close to the Weinberg method of quasi-particles [5]. Indeed, let the operator G,V have one EV 101~1 > 1. According to Weinberg [5], one should find g,,(E), the EF of the kernel G,V corresponding to OI,,, and then isolate the term -01~ 1g,)(g, 1 from the kernel; the remainder will then give rise to a convergent iterative series. It is much more difficult, however, to find g,(E) at positive energies than at negative, because in the former case the operator G,V is non-Hermitian (and unsymmetrizable). But in our approach one should know the function g, only at such negative energies E,, at which ol,,(E,,) = 1. This is sufficient to ensure convergence at small energies (and, very likely at any energy, as is suggested by example (a)). Moreover, for the series of iterations of G,V to converge it suffices, in fact, that y, contained in the projector r, coincide with the EF: g,(E,) at any energy E* < E, , i.e., be the EF of the Hamiltonian Ho + (l/a) 1xfor 01= ol(E,) < 1. In such a case, of course, the function 4 obtained as a result of the iterations will be different from $, but the relationship (48) enables

346

KUKULIN

AND POMERANTSEV

us to find $ if we know $ and e. These considerations permit us to give a recipe of a rapid convergence. When V < 0, one should take as yi all the bound states of the Hamiltonian Ho + /? V and the larger /I, the greater the number of terms in r and the faster convergence, since, with this choice the first N EV’s are eliminated. We have then that all Zi, < p-l. For pure repulsive potentials one should take the EF’s of the mirror Hamiltonian Ho - V as q’i, and, to improve the convergence, H,, - ,BV. Finally, in the case of potentials of uncertain sign, one should take as br the subspace pulled on the EF’s of both the direct and mirror Hamiltonians. Thus, in fhi.s context, our method of improving the convergence may be regarded as a kind of effective synthesis of the Feshbach projection technique and the Weinberg method of quasi-particles. It should be added that, with the increasing number of linearly independent vectors included in the projector J’, the approximations of the exact wavefunction $ by $,, and of f? by G, will be improving and, provided the dimensionality of the subspace CP is sufficient, $ N z&,, Z: N e, with a good accuracy [14]. In such a case, however, the convergence will, in general, be merely of power type. But if one includes only several main components and the remainder is iterated out, then the convergence will be already exponential [ 151. When the proposed scheme is used in practice, the functions of the bound states can be determined only approximately. Therefore, there arises the question whether the series of iterations of the kernel e,,Y will be convergent when the functions vi included in r are different from the exact EF’s of the Hamiltonian. It is clear that for sufficiently small deviations 6~~ from the exact EF’s of the Hamiltonian the convergence will take place, because the operator G,V and, therefore, its EV’s are continuous functionals of vi . Here we confine ourselves to the above remark and only wish to point out that, as is easy to show, the nearer to the threshold the bound state ‘pi, the larger may be the permissible deviations Sqi at which the series of iterations still remains convergent. We also note that the problem may be treated by fixing P)~as the bound states in the potential V, and then varying the potential by 6 Y. The particular case of 6 v = /3 V, has been considered above, as for the general case, we can repeat all that has been said about the variations 6~~ .

3. THREE-BODY

SCATTERING

In the system of three particles there are, along with the bound three-particle states, the bound states of separate pairs. For this reason, the orthogonal projection technique may be applied to project on both two- and three-body states. In the former case, the projectors are no longer finite dimensional. This, however, does not give rise to significant difficulties, as the aim of projection is the elimination of two-body states (and poles corresponding to them) from the two-body t-matrices contained in the kernel of the Faddeev equations. This problem was considered in [3], and the results obtained there will be described very briefly in Section 4. Another method of removing the poles of two-body t-matrices from the kernels of three-body equations

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347

in order to improve the convergence of iterative series has been developed by Fuda et al. [16] (see also [17]). Here we consider the three-body projection, aiming at improving the convergence of iterations of the Faddeev equations, as a direct extension of the proceding section to the various three-body systems; we also wish to indicate some possible advantages of applying such a projection in a number of problems. 3.1. Three DQferent Particles

Consider a system of three particles interacting through the pair potentials Vij = V, (ijk = 123,2!31, 312). The Hamiltonian of the system will be

H=H+

vl+ v3+ v,.

Let us apply the orthogonal projection technique suggested above to improve the convergence of iterations of the Faddeev equations (FE’s). The identity (21) for the GF leads, in a standard way [18].

e = Go+ G(l) + I32 + Q3), where @iI == G,V$,

(4%

to the modified FE’s :

fp) = ei - Go+ lQi(@) + (yk)),

(50)

and where Gi = (E - H, - V&l is the two-body GF, while all the GF’s marked with a tilde are defined by the relationship (13) via the corresponding unmarked GF’s. The system (50) differs from the ordinary system of FE’s only by the superscript tilde above all GF’s. In matrix notation the kernel of the system (50) is

Here the th.ree-row matrices c, P, p’, and 6 are diagonal and their elements are Gi , Vi , Ti , and C=i, respectively, while the elements of the diagonal matrix fare

Here r is the 3N-dimensional three-body projector, a, the numeral Hermitian matrix

a= and the matrix operator

0 1 1 10 I, 1 1 0

! 1

348

KUKULIN

AND

POMERANTSEV

represents the kernel of the ordinary system of the FE’s for the GF or the wavefunctions. The two-body t-matrices are defined in the usual manner Ti = Vi + ViGiViy GoTi = GiVi e

As is known [18], with the appropriate limitations on Vi the kernel Kis compact. We assure that it is so. In addition, the consideration to follow is concerned with the attractive potentials Vi < 0 and energies below the lowest two-body threshold E < E, .

Under these assumptions the kernel K is symmetrizable, since (-G,), (-G,), and consequently also (- Ti) (see [lo]) are positively determined. The symmetrization may be achieved, for example, as follows: K = G,(- p)1’2 K,(- If;)-“’ Go’, KS = -(-f’)W

&,,(-p)W,

(51)

K,t = K,.

The EV’s 01, of the operators K and K, coincide and, therefore they are real. The EF’s of K and KS are related in an obvious manner: I xn> = -G,(--)1’2

I xn’>,

1xn) = G,1(--)-1’2

1xns) = -G,lF-lG,l

j xn),

where 1xnS), 1xn) , and I j&) are the EF’s of the operators KS, K, and K+, respectively (they are the columns of three functions). The basis {xnS} may be considered complete and orthogonalized. Then the conditions of completeness and orthonormalization

hold also for the biorthogonal basis {xn , jjn}. Taking advantage of this basis, one may conveniently write the expansion of the operator K: K = c 01, I xn>
Now we consider the kernel of the modified system (50): (52)

where PO = r(rG,r)-l

FG,,

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THEORY

and

We note that the definition of the projected two-body t-matrix Fi given here is different than. that given in Section 2.2. By simple manipulations one readily obtains the following representation for ?‘: ?

=

-(-Q/Z

where i? = 1 - (-?)1;2 G,p(&G) operator inverse to R is:

A(-Q/2,

I?)-’ G,(-p)1/2.

A-l = 1 + (- F)liz G&p(-G,)

It can easily be seen that the

f)-l pG,( - ?-)l,'O

whence it follows that R is positively defined and II Z?11.< 1, i.e., for any p?there holds the inequality

Xv’I~-l i v> 3 (P’ I TJ>.

(53)

It is evident that the EV’s of the operator I? coincide with those of the operator ?A(1 - PO+)G, = -(-p)1/2

I?1/z(-~)1/2 (1 - I’,+) G,d.

Making further similarity transformations, we obtain the Hermitian operator I?, which is equivalent to I? in the orthogonal complement to its zero-space R, = --(I - $+) fi1/2(-7)1’”

G&4)1/2

j@P(l - f’) = (1 - p’) @/2&~W(l

- p)

where p = &l/2(-

59-l/2 po(- p/2

fill2

is the projector obtained from P,,by the same similarity transformation. In analogy with Section 2 from the minimax principle we deduce that the EV’s of the operator I?, iand, therefore, also those of R are real and satisfy the inequalities &nas < &ax,

amin 3 &in 1

where aR stands for the EV’s of the operator f?1/2K,1?1/2. Taking into account (53), we have for aR: a;,

= IlJy (cp I IZ1f2K,l?1i2 1 p.) = max((g, / k’,?) p?>/(p, ) R-l i 91)) < Et? (q j K, ’ F? y= ~max .

A similar inequality holds also for amin . Therefore,

350

KUKULIN

AND

POMERANTSEV

and, as a consequence, all EV’s are confined within the limits

As in the case of the one-particle problem, these inequalities hold for any projector. It will be noted that from (52) it follows that the operator R has the 3N-dimensional zero eigenvalue (the dimensionality of the projector P, is 3N). Repeating literally the arguments of Section 2 that in the case of the eigenstates projection the equality & = 1 is impossible we come to the conclusion about the convergency of iterative series for the modified FE’s (50), which is similar to the Theorem of Section 2:

If r includes all bound states of the three-body system (and only these states), the iterative series of the system (50) converges at E < EB provided that the EV’s of the kernel of the ordinary system of the FE’s satisfy the inequal& 01, > -1.

(54)

It will be recalled that, in contrast to the one-particle case this proof is valid only for the potentials of definite sign. Because the kernel Kincludes the matrix A, which is not of definite sign, the EV’s will necessarily be of different signs even in the case of definite-sign potentials. As the functions a,(E) are not, in general, monotonic [19, 201, one cannot replace (54) by a condition such as a,(E,) > -1. Consequently, the criterion we have arrived at is not easily verified. Large negative values of oin in the three-body system are less probable, however. In particular, Narodetsky et al. [20] have found that, in the case of separable pair potentials, the maximum (modulo) negative EV amin > -1. Moreover, for pure repulsive potentials it can be shown (see Appendix B) that in the general case of three different particles cy, > -2, which is an extension of the result obtained by Harms [21] for the system of identical particles with the separable potentials. 3.2. A System of Two Equations As is known, in some cases [19], the simultaneous Faddeev equations may be reduced to a system of two equations. For instance, for the total GF one may write the decomposition: G = Go + G’12’ + Gt3’,

(55)

where G’12’ = G,,V12G;

Gt3’ = G,(V,, + V3& G = G,,V3G.

The notation is somewhat changed here, as compared with that adopted in Section 3.1. For the components Go2) and Gt3) we get the system of equations ,312

= G12 - G, + G,,VuG’3’,

Gt3’ = G, - G, + G,V,G’l”‘,

(56)

SCATTERING

351

THEORY

where now G,, = (E - H,, - V&l,

G, = (E - HO - V&l.

Equations (56) are convenient when the GF G3 can be found somehow. Specifically, when the third particle has an infinite mass, which is a good approximation for electron scattering by atoms and nucleon scattering by nuclei, G, is simply the convolution of single-particle GF’s g,, and g,, . In principle, G, can be found from a system of equations similar to (56). Let us now consider the projection as applied to the system (56). Again, making use of the identity (21), we obtain a projected system which differs from (56) in that all its GF’s have the superscript tilde. The kernel of such a system will be

where ‘-(

Gl,~(I’Gl,~)-l o

r

0 GJ(I’G,T)-l

K2 = @A,

I’ 1’

(58)

is the kernel of (56). Then, again confining ourselves to the case E < Eb and Vi < 0, we conclude that in such a case the operators e, p, and p = P + PC% are negatively defined and, consequently, all EV’s of the kernel Kz are real. All further arguments of Section 3.1 are valid for the kernels (57) and (58). In particular all EV’s of the operator & , Z, < 1, if all the bound states are included in r. But the operators K, and l& , as any operators of the form (g t) m . g eneral, are characterized by their spectrum being symmetric about zero. Indeed, if (2) is the EF of such an operator belonging to the value 01, then (-2) is obviously the EF belonging to the value --a. Therefore, the projected operator II;, has all 1 gi, [ < 1 and, a consequence, the iterative series of the projected system (56) converges at E < E. These arguments are valid for purely repulsive potentials, but, as such a system does not contain the bound states, it always has ] 01~1 < 1 and the iterations of the system (56) are always convergent at E < Eb , no matter how strong are the potentials Vi . 3.3. The System of Three Identical Particles All infere:nces of Section 3.1 are of course applicable in this case, too. For the identical particles, however, the system of the FE’s reduces to a single equation, as the condition of identity requires that the wavefunction be symmetrical. We consider the case of bosons, in which only the symmetric solutions of the FE’s have a physical meaning. T.hese solutions can be obtained by symmetrization from the solutions of the equation + = @ + 2G,TP#, where P is the operator of particle permutation.

(59)

352

KUKULIN

AND

POMERANTSEV

The problem of finding EV’s for the matrix kernel Kalso reduces to a single equation 2G,TP

1g,:‘, = xi, 1 g,j

(60)

with three EV’s of the matrix kernel K for each EV an of the operator Kl = 2G,TP, namely, a single a, and a double -a,/2 to which two linearly independent “antisymmetrized” functions correspond (see, e.g., [22]). In much the same way, the projection of the FE’s (59) leads to the equation for EV’s: I;; 1g”,> = 8, 1grL where

It is meant here that the two-body operators labeled by the index i are expressed in their variables and, therefore, (61) is independent of the number i. In accordance with the principal result of Section 3.1, if r includes all bound states, i.e., both physically realized symmetric and antisymmetric relative to P, then for the projected kernel R all L?&< 1. But for the identical particles it means that a, > -2, because, along with the value &, there are also either -a/2 or -2& values. The same result may be obtained if we proceed from the kernels K1 and K1 . Then all calculations of Section 3.1 remain unchanged, the role of the matrix A being performed by the permutation operator P. But as not only unity-valued EV’s of the operator K1 , but also the values B, = -2 (these correspond to “ghost” states unobservable physically [20, 221 and antisymmetrical) correspond to the bound states of the system, the projected kernel x1 has no eigenvalue Zi, = 1 and 2, = -2. Hence follows the inequality 1 > Zi, > -2. It, however, does not supply us with a stronger condition of convergence of the iterative series and, therefore, the condition (54): a, > -1 remains, as before, the convergence criterion. The operator Kl as well as K is not sign defined. Even in the case of the s-wave separable potentials, when Eq. (59) reduces to a one-dimensional integra1 equation, each term of the expansion of the one-dimensional integral operator K1 in partial waves is not, in general, of fixed sign [19]. It seems, however, that there is a sufficiently broad class of potentials, in particular, separable ones, for which the sign fixedness of each term of the partial expansion of the operator K1 takes place. Jn this case it may be asserted that in the s-wave approximation the projected kernel iterations will be convergent. On the other hand, since any noneigenstate projection diminishes the maximum (modulo) EV of the kernel, having taken a sufficient number of appropriate terms in r, we can achieve a rapid convergence of the rearranged iteration series. The scheme of the method coincides then with the purely formal derivation in Section 2.4. which is valid for any many-body system. The inclusion of spin makes the symmetry properties of the identical particle system more complex. The physical and ghost states in the fermion systems will be examined in detail in one of our subsequent works. It will be only noted here that the

353

SCATTERING THEORY

systems of three and more nucleons comprise the maximum-spin channels in which the physically bound states are absent while the ghosts (forbidden by the Pauli principle) are present (for example, in the quartet channel of n-d scattering, in the quintet channel of d-d scattering etc). According to the above considerations, it is the such ghosts that give rise to the divergence of the iterative series of FE (and probably any N-body equations of the Yakubovsky type) and, therefore, an orthogonalization to the ghost should be made to obtain a convergent series. Moreover, our calculations have shown that such an orthogonalization gives the zero-order approximation of the elastic scattering amplitude (i.e., the pure orthogonal scattering) which is very close to the exact value, in constrast to the situation with the eigenprojection where the results are similar to the potential scattering (see Fig. 2 herein and also Ref. [6a]). Thus, it appears that there is a sufficiently wide range of application of the method of orthogonal projection of the FE’s in which the convergence of the modified iterative series takes place at least in the neighborhood of the two-body threshold.

4. OTHER APPLICATIONS

OF THE ORTHOGONAL

PROJECTION METHOD

4.1. Introduction of the Bound State Function into the Scattering Problem as a Part of the Hamiltonian

As was shown by Saito [7], and particularly by the authors of works [6], taking into account the orthogonalization condition (of type (l-2)) alone yields in many cases the phase shift similar to the exact value even without interactions. That is, the phase shift 8,(E) of the orthogonality scattering wavefunction & (26) is in many cases, rather close to the phase shift 6(E) of the exact solution (see numerical examples in references cited above). In a more realistic situation, the Hamiltonian is often known not quite completely, whereas the information about the wavefunctions of the bound states can be extracted from independent data (electromagnetic form factors, cross sections for direct nuclear reactions, etc.). In this case, applying the orthogonal projection technique described above, we can significantly reduce the uncertainty in the solution of the scattering problem by introducing additional information about the wavefunctions of the bound states in the projector r. This course will enable us, apart from other things, to separate the uncertainties in the discrete spectrum from those in the scattering problem. A good example of this kind is the description of the 3N-system on the basis of the realistic NN-potentials. It is well known [23] that the realistic NN-forces proposed so far can-satisfactorily describe neither the binding energy of 3H and 3He nuclei nor the wavefunctions for the ground states of these nuclei, which is evidenced, for example, by the unsatisfactory description of the experimentally found electromagnetic form factor for 3He or of the n-d doublet scattering length “a. It is possible, however, to find the wavefunction for 3H from independent data (from a fitof the experimental charge form factor, etc.) and introduce it into the scattering problem

354

KUKULIN

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POMERANTSEV

via the three-body projector r (see Section 3). Then we can again try to calculate za and the doublet phase shifts. The agreement with experiment will then testify to the decisive role of the correct wavefunction for 3H. A very similar situation obtains also for the 4Nsystem. The same formalism of orthogonal projection may be used in a problem of a different kind. Suppose we have to calculate the probability of the monopole electromagnetic transition in the three-body system, e.g.,

in the scattering channels with the same quantum numbers as in the ground state tion (y-quantum is then virtual). In calculations of this kind the ground state function is usually taken in the phenomenological form, while the scattering functions are sought by solving a dynamical Faddeev or Schrijdinger equation. If the calculation of the continuum wavefunctions leaves out the requirements of orthogonalization to the chosen ground state function, the functions for the initial and final states usually turn out to be nonorthogonal, which leads to spurious transitions. In [24] it is emphasized, in particular, that in the electrodisintegration of the 3He nucleus into the two- and three-body final states the monopole Coulomb transition COplays the decisive role at low energies. Therefore, it is highly important to take into account the orthogonality of the functions to the phenomenological ground state function. The method of orthogonal projection developed above enables us to include, easily and naturally, the additional orthogonality conditions in the wave equations to be solved and to obtain a rapidly converging iterative scheme for the solutions of these equations. 4.2. Taking Account of Pauli’s Principle in Direct Nuclear Reactions

In [3 J, the developed method of orthogonal projection was applied to the rearrangement of integral equations (of Lippmann-Schwinger and of Faddeev) describing scattering in the system of compound particles (d, t, 01,etc.). Such equations are very suitable in particular for the integral description of deuteron-induced direct nuclear reactions. The purpose of this application was to eliminate from the kernels of these integral equations the virtual transitions prohibited by the Pauli principle. Here we shall not repeat the detailed arguments of that work (see also [ZS), but recapitulate the main results (the same problem is discussed, but with an entirely different approach, in [26, 271). If the two-body states, which are forbidden by the Pauli principle in the twoparticle subsystem k (k = 1,2,3), are included in the two-body projector l-‘, (it will be recalled that is an infinite-dimensional projector), then, as was shown in [3], the “channel” three-body GF’s elk) satisfy the modified system of the FE’s G(i) = ci&(G,, + G(j) + G(k)), where the single-interaction

ijk = 123,231,312,

GF’s ci = (1 - G,,v&l G,, , while the pseudopotential

SCATTERING

355

THEORY

pi = Vi + hr is determined as usual. It can be shown quite easily that at X ---f co A is not contained in the kernels eipi and (cf. (30)) the new kernel has the form

where the Pauli term dKi = -r,(l + GiV.) is responsible for the prohibition transitions into the occupied states. If we take into account the Faddeev decomposition of the total GF

of

e = G, + e(l) + 1’32) + e(3), it can readily be shown that I’$,,, ---f 0, i.e., that the total modified GF does not contain components corresponding to the occuped states. If the modified scattering operator F is determined as usual

it is easy to d.erive the modified integral equations for the three-body wavefunctions i,@) = 8ik,@k, + G&@(j)

+ a,&))

where rPkOis the wavefunction for the initial state of the system corresponding to the scattering of the particle k, from the bound state of two other particles. We note that the two-body projectors r, included in the two-body pseudopotentials lead to a complete projection of the total three-body wavefunction, i.e., to the removal from it of the contribution due to the forbidden transitions. A similar rearrangement may be carrie:d out also for the Lippmann-Schwinger simultaneous integral equations describing the deuteron-induced nuclear reactions [3,25]. The general formalism for the treatment of the two composite particle scatterings on the basis of LSE for the A-particles involving the Pauli principle was developed in [26] with special emphasis on the deuteron-induced (elastic scattering, stripping) nuclear reactions. In contrast to [26], we treat the problems of this kind using the three-body model and, hence, the Faddeev reduction of the three-body Hamiltonian. In this case tlhe Pauli principle is approximately included, i.e., at the level of only the two-body projection. In fact, we derive the Faddeev equations for three composite particles where the two-body interaction is given by Saito’s orthogonality condition model which seems to be a proper physical model for the true complex nonlocal interaction of two composite particles. The discussion of important physical implications of the inclusion of Pauli’s principle in the amplitudes of the deuteron-induced nuclear reactions, in particular, the deuteron breakup in the nuclear field caused by the Pauli effect alone (i.e., in our terminology as a result of the dKi Pauli terms in the kernels of the equations), can be found in a number of papers [26,27]. In particular, it has been shown in [26] how the contribution ‘of the orthogonal scattering for the deuteron-induced nuclear reactions can made single.

356

KUKULIN

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4.3. Correction of the Shortcomings of the OP W Approach in the Theory of Solid State The purpose of this short paragraph is to indicate the possibility of using the suggested method of orthogonal projection instead of the well-known OPW (orthogonal plane waves) technique, which is widely utilized in calculations of the band structure in solids. It will be recalled that in solid state physics there is a variety of methods (pseudopotentials of Philips-Kleinman, of Austin, OPW method, etc.; an excellent exposition of these matters is available in [28]), whose aim is the explicit elimination of contribution from the occupied one-electron low-lying states in atoms of the crystal lattice in order to obtain the convergent Born series (or the series in the perturbation theory) for the energy eigenvalues b(K) of the Bloch states. It can be shown [28] that the problems arising in such a case are very close to the problem of convergence of Born series for the potential scattering, which is here under study. In particular, in the Phillips-Kleinman pseudopotential method one replaces the true atomic potential V(r) by the pseudopotential

where vi are EF’s of the low-lying states of an atomic core and Ei are the corresponding EV’s. It is clear, however, that at E > Ei the pseudopotential leads to the appearance of the known singularities in the continuum-positive energy bound states [29] and it is just those singularities that give rise to the Weinberg “pathological” trajectories ZEK with self-crossings and breaks near [30]. If, then, the Born series is convergent at E = 0 it may again diverge as the energy increases, etc. The orthogonal projection technique suggested above is free of all these shortcomings, because it shifts the singularities corresponding to the bound states not in the physical region, as in the case of FF,, but to infinity, thereby removing the singularities from the region of physical energies. As has been shown in Section 2 the trajectories Z(E) for the orthogonized kernels (e,v) run in a standard way and have no self-crossings. In some cases, the analytical pseudopotentials of the VP, type are replaced by the OPW basis I x:,;~> = I k + g> - BJ&

I k + g>,

where 1k + g) is the plane wave normalized to the unit volume; k is a fixed wave vector; {g} is the reciprocal-lattice vectors forming a discrete set; r = xi 1vi)(vi I is the operator of projection onto the bound core states; Bk = x,1 k + g)(k + g 1is the Bloch projection operator. If the overlapping of the bound state functions corresponding to the various lattice points is negligeable, then B,rB, = r, is the operator of projection onto the eigenfunctions Bkvi = vi,k of the total Hamiltonian with energy di. Such OPW basis has some shortcomings, in particular it is overfilled. Quite a number of procedures eliminating the shortcomings of the OPW basis have

SCATTERING THEORY

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been known1 [31, 321. In particular, Girardeau [31] proposed that some additional functions “resembling” the filled states of atomic core should be subtracted from the OPW-functions. Although this procedure eventually gives the complete orthonormalized basis, it looks somewhat artificial. The basis constructed of the eigenfunctions GCnof the pseudo Hamiltonian I?0 = H,, + hr, which satisfy the Bloch boundary conditions may prove to be much more handy. Because of the periodicity of the boundary conditions, the above said functions form a discrete set (at a fixed k) which is complete in a subspace orthogonal to r, (at h -+ co). The eigenvalues, E, , can be determined from the condition det(vi,k 1G,(E,)/ P)~,,J= 0. Quite an identical basis is used in work [32], the authors of which proceeded from the following pseudoHamiltonian inserted by Shakin et al. [6a]:

where Pk = B, - r, . The advantages of such a basis over OPW are discussed in [32].

5. CONCLUSION Let us briefly discuss some aspects of the practical application of the proposed scheme for the rearrangement of the Born series. In practical calculations within the framework of the orthogonal projection technique one should know the bound state wavefunctions for a given Hamiltonian H. These functions seem to be the most conveniently calculated by the variational method, where the basis functions can be chosen in the most suitable form for further use (for calculating integrals, etc.). In this approach, the many-body integral equations (difficult for numerical treatment) are used only where their application is the most effective, i.e., in describing many-body scattering, while reserving the discrete spectrum, which is simpler for the calculations, for variational methods. It should be noted that as Amado and Rubin [33] have recently emphasized, the characteristics of convergence of the iterative series for the many-body scattering are determined solely by the characteristics of the kernel of the equation and are independent of the process being considered (elastic scattering, inelastic scattering, or breakup). That is, after the orthogonal rearrangement, the three-body equations obtained may be used to calculate any three-body process (in the channel with the same quantum numbers). Further, as has been shown above, the developed method of orthogonal projection using the orthogonahzing pseudopotentials may be applied to solve diverse problems in the various fields of physics. There are, at the same time, other methods of solving many of these problems and different methods are being employed now to solve the different problems. 1 The present authors thank the referee for directing our attention to the works referred to.

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For example, many studies have dealt with the task of improving convergence of Born series in the two- and three-body scattering problems. Several effective methods have been developed so far, of which we should like to mention, in particular: (i) The Weinberg method of quasi-particles [5], which requires, however, the eigenvalue problem be solved many times, thereby consuming too much computer time. The approach developed by Ah ef al. [12] is intimately related to the Weinberg method. This approach proved to be quite efficient but it is labor-consuming because a system of three-body integral equations with separable potentials has to be initially solved with the subsequent use of the obtained solutions to carry out further iterations of the rest interaction. (ii) The Sasakawa method [34] and its various generalizations made by Austern 1351, Soper [36], Kowalski [17], and Sasakawa himself [37]. (iii) The method of subtracting the pole terms from the three body integral kernels, which has been developed by Fuda and Whiting [16]. In both approaches a more rapid convergence of Born series is achieved. Yet these approaches do not involve explicitly the bound states and, therefore, it is difficult to say a priori whether or not the series converge at low energies. (iv) The application of the Pade approximant technique to the calculation of three-body amplitudes by extending the sum of the Born series in the coupling constant, which has been realized by Malfliet and Tjon. This method is quite reliable and efficient but it requires a high accuracy to make a good extrapolation, i.e., a large number of iterations is necessary. (v) The application of the method of moments (developed by Vorob’ev) to calculations of the three-body scattering amplitude, as has been suggested by Harms [39]. Although these methods have effectively solved the problem of improving the convergence of iteration series, they, however, fail to include, for example, Pauli’s principle in the deuteron-induced three-body nuclear reactions. The latter problem has its own methods of solution [26, 271, which, unfortunately, are useless in the theory of Born series. Besides, in all the above-mentioned approaches aiming to accelerate the convergence of Born series it is difficult to take directly into account the information on the discrete part of spectrum for solving the scattering problem. In this connection, we believe that the pseudopotential orthogonal projection method will turn out to be a versatile tool for solving a number of scattering problems. On the basis of this method the authors intend to develop an efficient iteration scheme for calculating scattering in the problem of three and four bodies (n-d, GHe, d-d scattering, etc.), as well as for economical treatment of direct nuclear reactions, involving three particles.

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APPENDIX

A

Let us givle a more visual proof cf the main result of Section 2.2, confining ourselves for simplicity to the case V < 0, N = 1 (i.e., orthogonalization to a single function pl,,). Namely, we shall prove that if the function v,, involved in I’ is the only EF of the Hamiltonian H with the energy E,, -C 0, then the iteration series for the kernel f?(z) = Co(z) V converges at all energies z < 0. At the same time, some additional useful properties of the EV’s 8, of the kernel l? will become apparent. Let us take advantage of the expansion of the initial kernel K = G,,v in the EF’s I \g,j, which are given by @I (g&*>

I &z(z)> = 4)

I &2(4>,

(Al)

I G?G) I g&D

= --6mn 9

(A21

c I g,(z)‘(g&*)l II

= -Go(z).

643)

Since we are interested only in the real 2 < 0, in what follows the z-dependence will be omi.tted where possible. As is known [lo], in the case under consideration CX~> 0, dol,/dz > 0. As usual; we assume that the EV’s 01, are labeled in the order of decrease: 01,~~ > 01, . Writing the EF’s of the kernel I? as an expansion in hJ~ we obtain an equation for determining & in the form det a = det (an i

Z) a,, +

%PnE7L
(A41

where

Assuming that the infinite determinant in (A.4) converges, this is always so; in particular, if K is the kernel operator, i.e., C,“=, I 01, \ < co, it can readily be calculated as the limit of finite-rank determinants: G45)

Equation (AS) has been derived using the condition yields the relationship

obviously, 13 yk = 1.

of completeness (A3) which

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AND

POMERANTSEV

Thus, the equation for the EV’s of the operator fi takes on the form

Hence (for details see [4]): (9 (ii)

0 < k < ao, z. = 0.

One value is always zero which corresponds to orthogonalization single function y0 .

with respect to a

(iii) Since, in general, yn # 0 the remaining & do not coincide with the other EV’s CY,and may thus be determined from the equation (A7) (iv) If ynP1 > 0, yn > 0 in the interval (an, 01,-r) there is one and only one root of Eq. (A6) which we denote & . Thus we have the relationships

the sign of equality being possible if, and only if, yn-l = 0 and yn = 0, respectively. Up to now, y. has been an arbitrary normalized function. If, however, p. is the EF of the Hamiltonian with the EV En0 < 0, i.e., v. = NgnO(&J, the following properties of Zi, are easily established. At z = En , n # no, when a,(E,) = 1 i.e., at energies corresponding to the bound states yn different from yo, yn(En) = 0, as (g, ; 90) = N-l(v, 1 vo) = 0. Detailed analysis [4] reveals that at points z = E, the curve &(z) is externally tangent with respect to the curve an(z) at n > no, and is internally tangent at n < n, . Let us show that Eq. (A7) has no roots equal to unity for any Z. Since now (yO I(G;’ - Y) = (cpOI(z - Eno) then /%G) = (1 - G)(Z - En,)-* Therefore, taking into account Eq. (A3): m

= --B-l(z - E&l

# 0.

(~0

I GO’ 1gn:\.

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THEORY

At z = En0 tlhe equality & = 1 is impossible too, as yn”(En”) = W2B-l Thus, for the eigenstate projection follows (altering the labeling)

f 0.

the inequalities

(A8) should be rewritten

as

the sign of equality occurring at points z = E,, , n # n,, . And only at these points are the &&(=a,) equal to unity. Now if the system contains only one bound state yu with the energy E,, , then for any z G,(z) =/L1. Consequently S,(z) < 1

ifz
Co,

since &(z),-.-~ -+ 0. This proves the convergence of the iteration series of the kernel k We note that the above arguments can be readily extended to the sum of potentials of different sign and N > 1, but this also entails a considerable loss of visualization of the results.

APPENDIX

B

Proceeding from the physical considerations, Harms [21] suggested that at energies below threshold in the system of three identical particles there cannot be large negative EV’s of the .kernel Kl = 2G,TP generated by the repulsive part of the potential. This idea has been confirmed by his estimate for the case of the separable, pure repulsive potential (V, > 0):

Apropos of this inequality we should like to remark the following. In the system of identical particles with any Vi > 0 the bound states cannot exist, hence, taking into account what has been stated in Section 3.2, we immediately obtain -2

< 01, < 1.

It is then assumed, however, that there are neither physical nor “ghost” states. The latter assumption is of course unphysical.

032) bound

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We shall demonstrate here that in fact (Bl) is in no way connected with the identity of particles, but is a consequence of the mathematical structure of the kernel K of the system of FE’s for any positive potentials Vi > 0 and energies below threshold E < Ef . In contrast to the case considered in Section 3.1, the operators I;, F, and -G, are positive here. Taking this into account, we have, according to (51)

Because the quadratic form in (B3) is real

Here we have made use of the relationship P-rG-l = l%?-l and of the fact v-r is positive. Thus, we obtain the inequality (Bl). Assuming that the bound states are absent and that LY,are the continuous functions of E we get (B2). Once we have established (B2), coming back to the case of identical particles, we conclude that in fact thare are no “ghost” bound states in the system with Vi > 0, since from the inequalities (B2) for the EV’s of the matrix operator K there follow the same inequalities for the EV’s of the scalar operator KI .

ACKNOWLEDGMENTS The authors wish to express their deep gratitude to Professor L. D. Faddeev for stimulating discussions and support. We are also grateful to Professor V. G. Neudatchin for attracting our attention to a number of interesting problems touched upon in this paper and for helpful suggestions made in the course of this work.

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