Separable operator expansions for the t-matrix

Nuclear Physics A241 (1975) 429-442; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

SEPARABLE

OPERATOR

EXPANSIONS

SADHAN K. ADHIKARI

FOR THE

T-MATRIX

and I. H. SLOAN

Department of Applied Mathematics, University of New South Wales, Kensington, NS W 2033, Australia’ Received 24 July 1974 Abstract: A general scheme is developed for constructing finite-rank approximations

to the t-matrix. The emphasis is on obtaining approximations that are good in a particular context (specifically, in the kernel of few-body problems), rather than on attaining good pointwise accuracy. The scheme in its most general form makes use of two arbitrary sets of expansion functions, and various choices for these functions are considered, and their virtues assessed. Other expansion methods (Weinberg series, unitary pole expansion, and the expansion of Ernst, Shakin and Thaler) can be obtained from the formalism through special choices of the expansion functions, but the methods proposed here, unlike those methods, do not require the solution of an eigenvalue problem. Numerical calculations, using simple choices for the expansion functions, are carried out for the Reid ‘So soft-core N-N potential. They show that the convergence of the t-matrix expansion can be very good indeed, especially if it is used in the operator context for which it is designed.

1. Introduction

Even with the recent developments of the few-body problem in the last decade or so, it is still very difficult and complicated to solve the three-body problem with local potentials. Numerically, some of the problems can be avoided, and the dimension of the problem enormously reduced, if we can find a suitable rapidly convergent separable expansion of the two-body t-matrix. Much work has been done along these lines, and there are quite a few separable expansions already available I- 5). In all of the separable expansions so far proposed, emphasis has been on representing the t-matrix accurately in a pointwise sense. But that is not necessarily what is really wanted: usually what we really want is for the approximate t-matrix to be accurate when it is used in a particular context, for example, in the kernel of the three-body problem. The separable expansions described in this paper are specifically designed to be accurate in few-body contexts such as this. In this paper we mainly discuss separable expansions of the t-matrix, for the case of a local potential. Given the potential, we give a general prescription for constructing separable expansions for the potential, and hence for the t-matrix. The prescription is influenced, of course, by the particular way that the two-body t-matrix T enters into the few-body equations. The essential feature, for example of the Faddeev equations 6), is that T appears in the integral term in the combination + Supported in part by the Australian Research Grants Committee. 429

430

S. K. ADHIKARI

AND I. H. SLOAN

TG&, where Go is the free-particle Green function, which is singular if the energy is positive. On the other hand, the quantity II/, which is related to the three-body amplitude, is a smooth function of momentum provided that the disconnected parts of the three-body amplitudes have been subtracted off, which we shall assume has been done. Our aim therefore is to obtain a separable t-matrix that is accurate when it operates on Go times certain smooth functions. Our method is based on the recognition 7**) that one can explicitly construct a rank-N approximation to an operator, which is exact when it acts on a set of N predetermined functions. With a proper choice of expansion functions, it is possible to have a rapidly converging expansion, as well as an excellent rank-one approximation. Some previously suggested separable expansion methods ’ - 3, can be obtained from the formalism through special choices of expansion functions, as we show in the following section. Unlike those methods, however, the methods discussed here do not generally require the solution of any eigenvalue problems. The form factors in the present methods take the form of one-dimensional integrals, whose analytic properties can be studied. The analytic behaviour of a separable approximation for the t-matrix is very important if one wants to solve the three-body problem by the technique of contour rotation ‘). The contour rotation technique can be applied with the present expansions, and the form factors calculated along rotated contours. The separable expansion methods discussed here can be extended in an obvious way to more difficult few-body situations, and in fact that idea is the prime motivation for the present work. For example, the kernel of the four-body scattering equations contains the three-body amplitudes for each three-body subsystem, hence the four-body problem could be much simplified by obtaining the three-body amplitudes in a suitable finite-rank form, in just the same way as we are doing for two-body amplitudes in the present work. Of course, three-body amplitudes at positive energies are rather singular ‘- ‘I), and such a programme would be virtually doomed to failure if we were to insist on good pointwise convergence for the separable expansion. But the situation is much happier if the emphasis is rather on obtaining an approximation that is,good in a specified operator sense; and indeed it is already known that kernels with logarithmic singularities can be quite satisfactorily treated numerically with methods of the kind used here. In the present paper, however, we confine our attention just to the case of the two-body t-matrix. In sect. 2 we give the general theory of the method. In set. 3 we give numerical results for the simplest of our separable expansions, for the case of the Reid ‘*) ‘So soft-core potential. With a proper choice of expansion functions this gives a very good rank-one approximation, as well as excellent convergence at large N. Finally some conclusions are given in sect. 4.

SEPARABLE OPERATOR EXPANSIONS

43f

2. The seprabk expansions In this section we develop the separable expansions. The Lippmann-Schwinger equation for the two-body t-matrix T(E+&) has the form IT= V+VG,T = V+TG,I/;

(1)

G,(E) = l/(E - HO + ie),

(2)

where and where Ho is the kinetic energy, E is the total energy and V is the potential. The method is based on the fact ‘*8*3> that we can explicitly construct a rank-N approximate potential V, , witb the property of being exact when it operates on any linear combination of N selected functions lur>, . . ., IaN>. ft has the form

?I,m=l

where V’AiJI

= <%I%),

n, m = l,,..,

N,

(41

and where
&VI%) = Vn,), which can be easily The choice of the In the kernel of the is a smooth function

(5)

verified using eq. (4). functions I%> is determined by the context in which V operates. Lippmann-Schwinger equation (l& it operates on G&I’, where 2’ of momentum. Hence we choose the functions to be of the form

Is> = G,lL>,

n = l,...,N,

161

where the tfn) are smooth functions of momentum; more precisely, the V;) should be a good basis set for describing the momentum dependence of the t-matrix (or what is usually nearly equivalent, of the potential). With this choice, eqs. (3)-Q) become v, =

: ~G,I.O@,,<~,,zI, ?l,VZ=1

(7)

where VOnl,

= (v,lGolf,>,

v,G,lf,> = VG,tL>,

(8)

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

It is true that the expansion for I’, given by eq. (7) is energy dependent, but this is not a f~d~en~l objection, since the energy dependence arises directly from the

S. K. ADHIKARI

432

AND I. H. SLOAN

energy dependence of the context in which the expansion is to be used. In particular at positive energies the Green function GO is singular, and we believe that it is important in this case to take explicit account of the GO, as we have done above through eq. (6). For practical calculations, the expansion (7) is truncated at large enough N and used in the Lippmann-Schwinger equation (1) to solve for a separable t-matrix TN, TN =

i

C.-‘km,

VG,lf,Y,,(~A

n,m=l

where =


-

6,

SO)

W,MA

(11)

and where TN satisfies TN = V,+VNG,TN = V,+T,G,V

-

(12)

These are adequate formulae for practical purposes, but formally the structure of TN becomes clearer if we define a new set of functions Ih),

I4,) = If,> - J’%lO

(13)

In terms of these functions we have TN =

:

~G,lhhJ,,~~ntI~

(14)

n,m=l

where (J- ‘),,z. =
(13

These

equations are completely analogous to eqs. (7) and (g), and it follows that TN satisfies the analogue of eq. (9),

TNG,lk,) = TGJW,

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

W)

This is a nice property if we plan to use TN in the Faddeev equations 6), for there the f-matrix occurs in the combination TG&, where 9 is a smooth function if the threebody equation is properly arranged. The functions I/r,,)defined by eq. (13) are smooth functions so long as the V;J are smooth functions, Hence our TNis in fact exact when it operates on GO times certain smooth functions. The method outlined above actually includes a whole family of approximations, corresponding to different choices of the (u,l functions. We consider here a few possibilities, and discuss their virtues :

SEPARABLE

OPERATOR

EXPANSIONS

433

with eq. (6) replaced by 1%) = V;),

(%ll = bnllT; ,.

We)

values. Choice (17a) is the simplest. With this choice VN will be neither symmetric nor hermitian, and consequently the t-matrix will not satisfy the correct unitarity reiation. However, this choice has the great virtue of simplicity, since it requires one less integration than the others. Furthermore, it has been shown *) that the VNgiven by this choice is very effective indeed when used in the kernel of the LippmannSchwinger equation. (The moral can be drawn from this that the absence of exact symmetry is not necessarily a barrier to successful numerical calculations.) Choice (17b) makes the expression for the t-matrix symmetric, provided that the V;.) are real. In choice (17c), VNis explicitly .hermitian, hence the t-matrix obeys the correct unitarity relation. (The potential could be made symmetric as well as hermitian by replacing the G,, in choice (17c), and also the Go in eqs. (7) and (8) by the principal part of the Green function.) In choice (17d) the potential is real and hermitian and the t-matrix satisfies the correct unitarity relation, but the relation (6) has been abandoned. A detailed discussion of this choice with numerical results is given elsewhere 5). The final choice, wherepr

., pN are selected momentum

<%I = Q%ll?;

m=

l,...,N,

is the most difficult to implement, but is also perhaps the most interesting, since it leads to a TN with the property


m = l,...,N,

(18)

as can easily be verified using eqs. (14) and (15). (A related device has also been used by ref. 3b).) Thus
(%I =
(19)

434

S. K. ADHIKARI

AND I. H. SLOAN

where the functions on the right are eigenfunctions kernel, satisfying

The biorthogonality

of the Lippmann-Schwinger

relation ‘),

(21)

Ok,,lGol~,> = 4n,($,lWU then reduces eq. (10) to the Weinberg series

(22) The unitary pole expansion 2), which is similar to the Weinberg series but which uses only the eigenfunctions at the bound-state energy -B, is obtained in a similar way by choosing IQ

= GA-fW,(-W, (23)

<%A=
This case is discussed further in ref. 5). To get the expansion of Ernst, Shakin and Thaler 3a), the 1%) in eq. (3) are chosen to be eigenstates (either bound or continuum) of the full Hamiltonian Ho + V, and the (v,l are chosen to make the potential symmetric,

(%I = <%lK

n = l,...,N.

(24)

Finally, the Bateman 13) series for the potential is obtained by taking the 1%) in eq. (3) to the delta functions,

w4l) = @P--P,),

n = l,...,N,

(25)

together with eq. (24) to make vN symmetric.

3. Numerical results To see how the expansion works in practice, we calculations with the Reid ‘2) ‘So soft core potential, short distances, and which tits the ‘S,, nucleon-nucleon up to a laboratory energy of 350 MeV. In momentum of this potential is

have carried out numerical which is highly repulsive at phase shifts quite accurately space the S-wave projection

(26)

435

SEPARABLE OPERATOR EXPANSIONS

where fir = 0_7fm-‘, vI = - 10.463 MeV/fm3,

& = 4&,

& = 7/3,,

v2 = - 1650.6 MeV/fm3,

The partial-wave Lippmann-Schwinger

a3 = 6484.2 MeVffm3.

equation can be expressed explicitly as

s

mqP, P”)?;w’, p’; 4 p”‘dp”, 2 2ps - prr2 3-t

I;(P,P’; 3) = VP, p’)+

(27)

0

where V&J,p’) is the potential, T&, p' ; s) the off-shell t-matrix, p the reduced mass, and s = E+ is is the complex energy. The on-shell elastic phase shift ~5~ is defined by q(p,p:$

(28)

+i.> = - ieid1sin6,.

The actual numerical ~lculations were carried out in moments space, and in units h2/2p = 41.47 MeV/fm’. The calculations were performed with choice (17a) of sect. 2. With this choice the separable expansion for the t-matrix, eq. (lo), becomes N

TN= 1

(29)

~Gol.DJ,,(f,l,

where (J- ‘)m. = U,,l(Go

-Go

(30)

WJl.L>.

The basic functions fn(p) were chosen as in ref. ls) to be

utIlP> = at61> = f,(P) =

-&

$$I,

@2-l

n = I,. .

.)

N,

(31)

where G _ r is the Gegenbauer polynomial, a polynomial of degree (n - I), and where fl is a parameter which can be chosen for practical convenience. This set of basis functions is equivalent to the set (p2 +/32)-“, n = 1, . . ., N, but eq. (31) has the advantage numerically that the basis functions so defined are orthogonal on 0 5 p c 00 with a certain weight function, and this tends to make .I-‘, given by eq. (30), well conditioned. After a small amount of experimentation we found that /? = 4 fm- ’ gave best convergence in this context, and this value of /I was used throughout the calculations. To find T&P, p’; s) from eq. (29) we must evaluate the matrix elements , and also the matrix J-r defined by eq, (30). Their explicit form is

(32)

436

S. K. ADHIKARI AND I. H. SLOAN

The integrals were evaluated numerically by separating the &function and principal value parts, and then performing the principal value integration by an even-order Gaussian quadrature symmetrically located about the pole 16). The double integration in eq. (33) is helped by noting that the q-integration has already been performed in eq. (32). The fust results we give are for the scattering length, which with our no~alisatio~ is nothing but T&O, 0; 0). The scattering length obtained from eq. (29) for different values of N are given in the centre column of table 1. We see that it takes about nine terms ot get a reasonable accurate result. TABLE I Scattering length (in fm) N

Eq. (29)

Eq. (34)

1 2 3 4 5 6 7 8 9

+0.3 f 0.9 -13.1 - 14.9 -15.0 - 16.9 - 16.9 - 16.9 -17.1

-1.0 -1.5 -14.1 -15.2 - 16.0 -17.1 -17.1 - 17.2 - 17.2

But actually it is not very sensible for us to look directly at the momentum space matrix elements of TN,because TN was designed it to be a good operator when it acts on Go times certain smooth functions, not necessarily to be good in a pointwise sense. To test it in a more appropriate way, we use TN in the right hand side of the transposed Lippma~-~hwinger equation T = V+TG,l(

and so obtain a new approximation

to T,

We now claim that the pointwise accuracy of the t-matrix values obtained from this expression is a fair measure of the effectiveness of TN in the operator context for which it was designed. For most of the remainder of this paper, the quoted t-matrix values (unless otherwise stated) were obtained from eq. (34), i.e. they arise from using TN in an appropriate operator context. Now we recalculate the scattering length TO(O,0; 0) using eq. (34), and put these numbers into the last column of table 1. Clearly the convergence is much faster than with the previous pointwise use of TN,the numbers now settling down to the con-

I

4

I

p(fm-9

(b) - new

basis 1

Fig. 1. Half-off-shell values of the zero energy t-matrix, obtained from eq. (34) with (a) the old basis and (b) the new basis.

P (fm-1)

I

(4 -4

-2

0

i+‘)

POm-9 Fig. 2. Off-shell zero energy t-matrix elements T(1.4,~; 0), obtained from eq. (34) with (a) the old basis and (b) the new basis.

20 c I-

2

4

2

p

( fd

4

0

2

P @a3

4

Fig. 3. Real part of the off-shell r-matrix T(0.6,~; E+ iz) at _Etrb= 40 MeV, obtained from eq. (34) with (a) the old basis and (b) the new basis.

0

I

2 P (fm-1)

I

I 4

0

I

P @a.‘)

2

I

4

Fig. 4. Imaginary part of the off-shell t-matrix T(0.6,~; E+ in) at I&,, = 40 MeV, obtained from eq. (34) with (a) the old basis and (b) the new basis.

0

SEPARABLE

P (fm-1

OPERATOR

EXPANSIONS

439

p (fm-9

Fig. 5. Values of the t-matrix T(O,p; E) at E = EC.,, = -25 MeV, obtained from eq. (34) with (a) the old basis and (b) the new basis.

verged result after only about six terms, and thus supports the notion that IN should be more effective in an operator sense than in a pointwise sense. Before going on to other numerical results, it is worth remembering that the approximations (29) and (34) used in table 1 were obtained in a quite different way in ref. 15): there the latter appears as a variational result, and the former as a corresponding non-variational result, thus the superiority of the one over the other is given a variational interpretation in that work. Yet another derivation of the approximation (34) is given in ref. 8). There the approximation (34) is obtained by using VN, eq. (7), (with (u,l = Cf,l)in the kernel, but not in the inhomogeneous term, of the Lippmann-Schwinger equation (1); whereas the approximation (29) is of course obtained by using V, in both. The comparison in table 1 therefore draws attention to the fact that a unite-rank approximation may be satisfactory for use in the kernel, without necessarily being so satisfactory for use in the inhomogenous term. We now look at further numerical results obtained with eq. (34). Fig. la shows half off-shell values of the zero energy t-matrix for several values of IV. Fig. 2a shows some fully off-shell t-matrix elements at the same energy. Fig. 3a and 4a show the real and imaginary parts of off-shell t-matrix elements at a laboratory energy of 40 MeV. Fig. 5a shows off-shell t-matrix elements at a c.m. energy of - 25 MeV. Evidently the convergence in all cases is good. On the other hand, it is clear that the approximations obtained by truncating at N = 1 or 2 can give rather poor results, especially at zero energy. It is therefore worthwhile considering how the approxi-

440

S. K. ADHIKARI

AND I. H. SLOAN

mations for small Nmight be improved. It turns out that we can enormously improve the small N results in a quite simple way, by taking a new choice of basis functions to replace eq. (31), and the remaining part of this section is devoted to studying this question of a new basis. We note, first, an important qualitative reason for the poor accuracy of TN for small values of N, particularly at zero energy, namely that the exact T-matrix has an antibound-state pole situated very close to zero energy I’). (This pole is of course reflected in the large negative values of the scattering length.) Clearly, a successful approximation will have to reproduce this pole; and it will have to do so rather accurately, since even a small error in the position of the pole will lead to large errors in the t-matrix. With this in mind, we choose the new basis set so that the half-shell t-matrix at zero-energy is automatically exact. Specifically, the new basis functions 7” are defined by
n = 2,. . -, N.

(35) (36)

It then follows, by the same argument used to prove eq. (18), that (p’ = OlT,(E = 0) = (p’ = OIT(E = 0), for any value of N, i.e. that the half-on-shell TN is exact at zero energy. (In fact, eq. (35) is essentially just choice (17e), made here at the fixed energy E = 0.) The practical implementation of the new basis set is in fact quite easy, if we make use of the calculations that have already been carried out at E = 0 with the original basis set. In fact, if we replace the exact t-matrix in eq. (35) by TN’, given by eq. (29), where N’ is sufficiently large that TN, has essentially converged, then eq. (35) is replaced by


jl:C,,
i?l=l

(37)

where the coefficients (which are already available from the calculations with the original basis) are C, =

F(p’

= OlVG,lf,)J;;‘.

(38)

n=l

(All quantities in this equation are evaluated at zero energy.) Thus the new basis function J; is given quite explicitly by eq. (37) as a linear combination of the old basis functions fn. In practice we used N’ = 6. We now show numerical results [still using eq. (34)] obtained with the new basis functions 7”. Fig. 1b shows the half-shell t-matrix at zero energy : this of course is now exact, by construction, for all values of N. The interesting question is whether the

SEPARABLE

OPEUTOR

441

EXPANSIONS

same change of basis is helpful at other momenta and other energies. [The poledominance argument i7) suggests that it should be at least near zero energy.] Fig. 2b shows the same fully-off-shell zero energy t-matrix elements as fig. 2a. Figs. 3b and 4b show the same 40 MeV t-matrix elements as figs. 3a and 4a. Fig. 5b shows the same negative energy t-matrix elements as fig. 5a. These results do indeed all find the new basis to be much superior to the old for small values of A? In pa~icular, the rank-one result is now reasonable, and the rank-two result is good, in every case considered. TABLE2 Real and imaginary parts of the phase shifts (in deg), obtained with the new basis Real

~magina~

N eq.

(29)

eq. (34)

eq. (29)

eq. (34)

40

1 2 3 4 5 6 7 8 9

50.6 43.0 41.3 41.1 41.1 41.4 41.9 42.2 42.0

43.5 42.0 42.1 42.2 42.2 42.2

+2.2 -6.8 -3.8 -2.0 -1.0 -0.3 0 0 + 0.2

-1.0 -0.8 -0.3 -0.1 0 0

140

1 2 3 4 5 6 7 8 9

23.7 19.6 12.4 13.0 14.0 15.2 15.8 15.7 16.0

18.7 16.8 15.7 15.8 15.8 15.8

-0.2 - 14.2 -6.0 -1.5 +0.3 t-o.7 0 -0.3 t-o.0

-0.4 -3.3 -0.8 -0.2 -0.1 0

0 - 18.6 -6.5 +0.2 +l.O io.7 -0.1 + 0.2 -0.3

+0.1 -6.6 -1.5 -0.2 -0.1 0

280

+I.6 + 4.8 - 10.7 -7.1 -5.3 -4.5 -4.9 -5.1 -4.6

-0.8 0 -5.0 -4.7 -4.7 -4.7

Finally, in table 2 we give the real and imaginary parts of the phase shift at several energies. For completeness, the results are given not only for eq. (34), i.e. for the operator usage of TN, but also for the direct pointwise calculation using eq. (29). We have already remarked in connection with table 1 that TN is much more

442

S. K. ADH’IKARI

AND

I. H. SLOAN

accurate when used in an appropriate operator context than it is in the pointwise sense, and this is now brought out even more strikingly in table 2. At the higher energies some decline is evident in the quality of the low rank approximations, presumably because the basis set becomes less suitable than at the lower energies, but the overall performance remains reasonable. 4. Conclusion The numerical calculations of the previous section were carried out with the simplest of the separable expansion methods developed in sect. 2 [choice (17a)], for which case the rank-N approximation TN is given explicitly by eq. (29). The final results - figs. lb-5b and table 2 - show that the approximation can be good even for small values of N, provided that TN is used in an appropriate operator context, and that a suitable set of expansion functions is obtained. A direct procedure for improving the expansion set, starting from a rather arbitrary initial choice, is indicated in the previous section. As discussed in the introduction, we believe that the methods developed here for the two-body t-matrix will also be useful in more difficult situations, for example for handling the three-body amplitudes that occur in the kernel of the four-body problem, The present work provides a foundation for future work in that direction.

References 1) R. Weinberg, Phys. Rev. 131 (1963) 440 2) E. Harms, Phys. Rev. Cl (1970) 1667 3a) D. J. Ernst, C. M. Shakin and R. M. Thaler, Phys. Rev. CS (1973) 46, 2056; 3b) D. J. Ernst, C. M. Shakin and R. M. Thaler, Phys. Rev. C9 (1974) 1780 4) M. G. Fuda, Phys. Rev. 186 (1969) 1078; T. A. Osborn, Nucl. Phys. Al38 (1969) 305 5) S. K. Adhikari, Phys. Rev. Cl0 (1974) 1623; S. K. Adhikari and I. H. Sloan, Phys. Rev. Cl1 (1975) 6) L. D. Feddeev, ZhETF (USSR) 39 (1960) 1459; JETP (Sov. Phys.) 12 (1961) 1014 7) I. H. Sloan, B. J. Burn and N. Datyner, to be published 8) I. H. Sloan and S. K. Adhikari, Nucl. Phys. A235 (1974) 352 9) R. Aaron and R. D. Amado, Phys. Rev. 150 (1966) 857 10) S. K. Adhikari and R. D. Amado, Phys. Rev. D9 (1974) 1467 11) T. J. Brady and I. H. Sloan, Phys. Rev. C9 (1974) 4 12) R. V. Reid, Ann. of Phys. 50 (1968) 411 13) H. Bateman, Proc. Roy. Sot. Al00 (1922) 441 14) K. L. Kowalski, Phys. Rev. Lett. 15 (1965) 798; H. P. Noyes, Phys. Rev. Lett. 15 (1965) 538 15) I. H. Sloan and T. J. Brady, Phys. Rev. C6 (1972) 701 16) I. H. Sloan, J. Computational Phys. 3 (1968) 332 17) C. Lovelace, Phys. Rev. 135 (1964) B1225