Expansion of continuum functions on resonance wave functions and amplitudes

Nuclear Physics A309 (1978) 381-421 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

EXPANSION OF CONTINUUM FUNCTIONS ON RESONANCE WAVE FUNCTIONS AND A M P L I T U D E S J. BANG

The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark and F. A. GAREEV, M. H. GIZZATKULOV and S. A. GONCHAROV

Joint Institute for Nuclear Research, Dubna, USSR Received 6 March 1978 (Revised 24 May 1978) Abstract: The pole expansion (Mittag-Leffier expansion) of wave functions, scattering amplitudes, and

Green functions at positive energies are discussed in a mathematically rigorous way. The general proofs of convergence are supplemented by numerical calculations, which, for a simple example, show the convergence to be fast. Applications of the method to nuclear structure calculations are discussed.

I. Introduction

In nuclear physics, the problem of treating states with positive energy on an equal footing with bound states has for a long time drawn the attention of several authors 1). We are here specially thinking of the single particle states used as a basis in, e.g., shell model calculations or more approximate many body descriptions of nuclear structure. In such calculations, the continuum states must be included in order to have a complete expansion basis. The necessity of this has recently become more urgent, due to the growing interest in studying the giant resonances and other highly excited states, where the continuum admixtures are particularly important. Apart from the more conventional problem of single particle scattering, these states also play a direct role as final states or intermediate states in many nuclear reactions, like (d, p) processes, etc. The fact that the spectrum of positive energy states is continuous, and the states not square integrable, gives rise to a number of problems when these states are used as basis states in realistic calculations of the above-mentioned type. Therefore, in such calculations, the single particle potential is often approximated by one which has only a discrete spectrum, like the harmonic oscillator or the infinite square well, but the unphysical character of these potentials leads to complications connected with bad convergence of the expansion in certain regions of space. In general we can say that if a physical problem is solved by the method of expansion in a complete set, 381

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the convergence will be best if the basis functions, or at least some of them, are similar to the physical function, i.e. if they are solutions of an equation of similar type, with similar boundary conditions. A number of ways have been proposed to construct discrete sets of wave functions, which give uniform convergence of the expansion when certain conditions are fulfilled. The basis of Eisenbud and Wigner 2), however, shares the property of unphysical boundary conditions with those mentioned above. The basis functions of Kapur and Peierls 3) and the Weinberg 4, 5) functions form complete sets for positive energies and the Sturm-Liouville functions give absolute and uniform (or stronger than uniform) convergence for expansion of square integrable functions 6.7). In contrast to the bound states these discrete sets present the inconvenience that they are energy dependent. In reaction calculations, this may lead to some advantages, but in the solution of a physical problem by expansion in a complete set, it is desirable that only the expansion coefficients contain the energy dependence. A natural generalisation of the discrete bound state solutions of the Schr6dinger equation to positive (or rather complex) energy values was introduced by Gamow in the description of or-decay 8). Like the bound states, these states have energies which correspond to poles of the S-matrix. A representation of the S-matrix by means of its poles was given by Peierls 9). A theory of nuclear reactions, in which these poles are used for expansion of the reaction amplitudes, was developed by Humbler and Rosenfeld 10). Since the Gamow functions (which in the following we shall call the resonance functions) are increasing exponentially with r, the usual definitions of normalization, orthogonality and completeness are not valid for these functions. This is, to our mind, the main reason why they have not become more widely used. Lately, a number of methods of normalization of the resonance wave functions were suggested, all of which, however, are equivalent to the method of Zel'dovich 11-15). Recently, techniques for expansion of continuum wave functions on resonance functions were developed, using the analytic properties of Green functions. But in this approach it is in general difficult to answer the question of convergence of this expansion. It is in this connection an important observation that in the study of the analytic properties of the S-matrix, and of its expansion in pole terms, the theorem of MittagLeffler 16) played a decisive role, guaranteeing that the series converges uniformly inside an arbitrary closed contour, which contains no poles. It is therefore tempting to use this theorem also for the expansion of the wave functions of the continuous spectrum, and of the Green functions. We are here using the expressions obtained in this way for comparison with exact solutions in some concrete cases, following the work of Serdobolskij J T)..in some of the works, where Gamow states are introduced, they are used in matrix elements, say, for calculation of transition amplitudes or expansion coefficients, involving integrations over all space. The exponential growth with r of these states is then

POLE EXPANSION

383

neutralized by introduction of a regularisation factor. Although in some cases, as e.g. with the normalization integral 11), this procedure is unambiguous, we shall here stick to the more conservative idea of using the resonant states as a basis for the calculation of such matrix elements, where we need only the properties of the wave functions inside a finite radius, R. We, therefore, also, like many of the above-mentioned authors, only need to prove the expansion theorems we are going to use for this finite region.

2. T h e M i t t a g - L e f f l e r

expansion

2.1. THE MITTAG-LEFFLER EXPANSION OF THE S-MATRIX We shall here, and in the following, assume that our single particle potential V(r) is spherical, identically zero for r > R, and has no singularities (except for a numerical example, where a 8-function potential is used). These conditions are stronger than necessary to prove the theorems used in the following, but reasonable from the point of view that we are concerned with nuclear probelms. The Coulomb potential, as met in atomic and molecular physics, is hereby excluded, but the repulsive Coulomb force important for protons and e-particles is treated below. With the above restrictions, the Jost functions f(k) are holomorphic in the whole k-plane. We shall further assume that they have only simple zeroes in the k-plane. In the appendix we throw" some light on this assumption. For Sz(k) (= f l ( - k)/fl(k)), we can now, introducing

FR(k) -

St(k ) - 1 (h~_(kR))2 ' 2ik

(2.1)

prove that

FR(k) < constlkcl p° f o r p large closedcontour with Ikl > Ikc]which does not come near to any pole in the k-plane 16) (such contours have been proved to exist). In the above expressions, hz+ are Ricatti-Hankel functions of the first sort, defined as in ref. 18) and P0 is a non-negative integer. For a potential, of which the ruth derivative at r = R is different from 0, it can be proved that Po --< m + 2. Let us denote the poles of F R by k,, and the corresponding residue by r r The Mittag-Leffler theorem then states that if the poles are ordered according to increasing (or non-decreasing) values of ]kn], the expansion ~. F~(0) + q= o

rn ~= 1 \~,]

(2.2)

k - k,

P>-Po

converges uniformly in the k-plane within any closed contour which does not contain poles. So, expressing S by (I),

St(k ) = 1 + 2ik(h~-(kR))- 2FR(k),

(2.3)

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we have the different Mittag-Leffier expansions of S. A Mittag-Leffier expansion contains in general an entire function and a series of pole terms. This division is however not unique, it can be changed by a re-ordering of the expansion, as exemplified by the choice of p above. The conditions p > P0 > 0 are sufficient conditions, but not necessary; in the example of a 6-potential, given below, p = - 1 gives a convergent series. The advantage of the presentation of the Mittag-Leffler expansion given above is that the entire function is given for any value ofp. The pole series itself will, of course, converge faster for large p-values, as we shall see in some concrete examples below. It should be noted that since St(k) - 1 = O(k zz+ 1) for k --, 0 (except for the case of a bound state at zero energy), then in general FR(0 ) :p 0, since h~-(kR) = O(k -t) for k ~ O. 2.2. THE MITTAG-LEFFLER EXPANSION OF THE WAVE FUNCTION

In order to apply the expansion of the S-matrix obtained above in the most direct way to the scattering states, let us introduce the function e-½ira

~ So ~

k h~(kR)¢l+~(r)"

(2.4)

satisfies the radial Schr6dinger equation and for all r < R, we have d2 dr2 @G+

(

1(1+ 1) k2

V(r)) ¢tG =

r2

O,

(2.5)

\

and for all r => R, we have ~

-

h~+ (kR)e- v ~ {. St(k)- 1 "~ k ~J'(kr)+ - - - - 2i h~ (kr)) ,

(2.6)

where Jl are Riccatti-Bessel functions, defined as in ref. 18). The regular solution of the radial Schr6dinger equation occurs in the literature under different names, corresponding to different k-dependent normalizations. Our reason to introduce still one more variant, our qJ~, will become clear in the following. Now, at the poles k, of the S-matrix, the wave function also has poles. Its residues must be wave functions, which satisfy the same Schr6dinger equation as q/~ (or 0(+)), for k = k,, but for r > R become proportional to ht+, so that lim (k - k . ) ~ ( k , r) =

kok~

c.~o.(r),

(2.7)

where the q~.(k., r) are solutions of the eigenvalue problem

dE

dr2 q~.(r)+

( kZ. l(l+ r z l)

v(r))rp,(r) = O,

(2.8)

POLE EXPANSION

385

~o.(0) = 0,

(2.8a)

h~ (k.R)~r~o.(r)l,=R = ~o.(R)k.h+'(k.R)

+'(x) -

h+(x) .

(2.8b)

The constant c. must depend on the normalization of the ~o. These functions are identical to the bound states for Re(k,) = 0, Im (k,) > 0. For Re(k,) = 0, Im (k,) < 0 they are antibound states, and for complex k. values resonant states, among these the G a m o w states for Re (k.) > 0. The normalisation of these functions is naturally some generalisation of the normalisation prescription for bound states. Keeping in mind that we only want to use the properties of the wave functions for r < R, we shall use the normalization

fo'dr~p~(r)+q~Z"(R)[d(kh~'(kR)~l dk ~/Jk=k.

= 1.

(2.9)

In that case

h k2 . 2 2m

- E . - ½ir.,

where F. is called the width of the resonance [see the appendix, and e.g. ref. 10) for some relations between F, and ~o,]. Note, however, that this is identical to the observed width in the limit of Im (k.) ~ 0 [ref. 22)]. It is also a condition for the cross section to show any resonant behaviour at all, that Im (k,) < Ik.-k,.I (all m). This has for example the advantage of giving the same first order expression for resonance energy changes in the continuum as in bound states. Now, the constant c. is determined to be -- e-

c, -

½i~l

2k.

q~,(R).

(2.10)

A detailed derivation of c. for the case l = 0 is given below. Once the residues are given, the Mittag-Leffler expansions of the function ~,~ can be constructed:

• ki o0 {k']P+lc,q~.(r) ~,p(k,r) = ,:o ~ O?(O'r)'°i!. +.~=,= - -\l~.J k - k , '

(2.11)

for p = l = 0, this series coincides, apart from the normalization, with the formula of ref. 17). That the ~ can be represented by a Mittag-Leffler expansion is perhaps most easily shown by considering the relation between ¢ p and the Green function

G + = - k - l ~ ' ( k , r < ) f t ( k , r > ) e -½i"' ~ ( k , r) = - G+(k, r, R)

[fz(k,r >= R) = h~-(kr)], r < R.

(2.12)

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386

So, the expansion properties of 0~ are given by those of G ÷, which we shall study in the next section. It should be noted, that the advantage of presenting the scattering wave function as a sum of energy-independent functions, multiplied with energy-dependent coefficients is shared by all the Mittag-Leffler expansions with different p. The choice of p is to a large degree a matter of convenience, since already for p = - 1, the expansion will often be convergent, but it means, that the convergence can be improved, at the cost of ~nlarging the number of functions ~,a(0, r)(°. These are found as solutions, with the proper boundary conditions of differential equations, derived from the Schr6dinger equation. The equations for i = 0 and i = 1 are, e.g., both d2

dr e

tptG(O' r)(i ) _ ~{ lr(-lT+- -1) + V(r) ) OP(O, r) "),

(2.13)

and the boundary conditions OtG(O, 0) {°) = 0~(0, 0){1) = 0,

e½i'z'l'Gto R) {°) F(O)+ 'ez ~ , =

R 21+

1 '

e~i't~b~(0, R) {1) = RF'(O) + iRZfzo.

(2.13a) (2.13b)

(2.13c)

2.3. THE MITTAG-LEFFLER EXPANSION OF THE GREEN FUNCTION: COMPLETENESS RELATION The Mittag-Leffler expansion of the Green function was considered by several authors 23, 26). The simplest way to obtain such an expansion is presumably to start with the general expression for the radial equation Green function G + :

G;(k,r,r')=

~

n {bound)

~o.(r)q~.(r ') 1 ~ dk' q~,+( k ,' r)O~*(k', r') k2 2 + j_~, k+_k , -- k n

IE

k'

(k + = k + i T , 7 --+ 0 + ) .

(2.14)

The integral may be calculated using Cauchy's theorem, by closing the contour in the lower half-plane for such terms where that integral becomes small and in the upper half-plane (counter clockwise) for the other terms, each time using a large contour that does not come near to any pole. That such contours exist, was already mentioned above. Now the contributions from the poles with positive imaginary k-value will partly cancel the bound state part of Gz+, and the contribution from k + (7 ~ 0 +) vanishes,

POLE EXPANSION

387

leaving us with

G~-(k, r, r') =

tp.(r)tp.(r') 2k.(k-k.)

~

(r, r' < R),

(2.15)

n (all p o l e s )

where the ~p.(r) are the resonance functions, with the same normalization which was introduced earlier. We see that the Green function, for the case considered, has a Mittag-Leffler expansion with p -- - 1 . The same must therefore be the case for ~bl+(r < R). Furthermore, the Green function for k ---, ~ must behave as f i ( r - r ' ) / k 2 + O(k-3). F r o m this and (2.15) one concludes

½ ~ ~p,(r)~o.(r') = 6 ( r - r ' ) ,

r, r' < R,

(2.16)

n

q~.(r)~o.(r')/k. = O,

r, r' < R.

(2.17)

n

Eqs. (2.16) and (2.17), as well as the expressions for the Green function, should be read in an operator sense, i.e. K(r, r') is represented by ~.f.(r)f.(r'), if

g g i = ~ f~(r)

dr' f,(r')gi(r')dr'

is a convergent series• When the relations are true for all gi in a complete set for the interval 0 < r < R, one is allowed to write Ge(k , r, r') = ( ~ ( r - r ' ) / k ~ + O ( k - 3 ) , etc. That such a set exists, namely gi chosen as the eigenfunctions in an infinite well of radius R was proved by Romo 26). The asymptotic expression for the Green function is independent of R, and thus more general than (2.16) and (2.17), cf. also ref. 16). The first equation can be stated as an overcompleteness relation. Note, that the presence of the factor ½ is not merely a question of normalization, ;since it appears with the bound states also, where, for large values of R, this plays no role. The second relation of course also indicate a very high degree of linear dependence among the q0,. The fact, that we have completeness only within a limited volume may seem a drawback, but it is related to the large difficulties encountered, if one wants to use the ~o as an expansion basis in all space. The regularisation method tries to overcome these difficulties, but if the innet product of two state vectors is defined with that method, the linear dependence (2.17) is conserved, but the completeness is lost, even at the point where it is most needed, as seen from the following example, with l=0. Forr>R eikR

~,~(k, r) = T~k (S(k)ei*" - e- ,k,),

(2.18)

tp.(r) = tp.( R )eik"(~- R(

(2.19)

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J. BANG

et al.

So for the regularized overlap integral between ~oc and go. we get

tk~(r)go.(r)e-~r2dr

lim e~O

=

O~(r)go"(r)dr+go"( ) 2ik~lk

dr(S(k)e"k"+k)r--e"k~-k)~)e-~r2(e--+ O)

= f~qJ~go.dr- go.(R) e"k-k")R(s(k)e"k"+kR' \2ik i(k.+ k)

= =

e"k"-k'g']~k.--k)/

fo~ (i S(k)e2ik"-I 1 ~9~go.dr+ go.(R) k. + k 2ik + 2k(k. + k)

1

))

2k(k.- k

0oa(R)- -k,-2 k2 ,

~oGgo.d,-+ go(R)

(2.20)

but this is zero, according to (2.28) below. The expression for the Green function, given above, may also, using (2.17), be written in a n u m b e r of other ways, which may be useful in different contexts,

go.(r)go.(r') +½ ~. k go.(r)go.(r') G+ = ½Z. k2_k 2 . k. kZ-k 2

(2.21)

Here, the first term, which is even in k, represents the real part of the Green function, whereas the second, odd, term represents the imaginary part of G + (for real k). Another interesting formula is

G+

= ~ go.(r)go.(r') ~ go.(r)go.(r') ~, k 2 kff. +½ k.(k+k,)"

(2.22)

Here, the first term contains the poles, whereas the second term is regular at the points k = k.. For a sufficiently narrow resonance, i.e. if k. = ~c. + i7., so that [y.[ is sufficiently small, the pole term gives the usual resonance expression for 0 + :

+ ~

go.(r)go.(R) n2__),2. _ 2iK.7. ) •

(2.23)

k 2 ,~K

2.4. THE RELATIONS BETWEEN ~/,C;(k,r) AND q~,(r): NORMALIZATION

In this section, we shall for simplicity consider the case of I = 0 only. We shall, for this case, derive some useful relations between qjG and go.. For / = 0, we have, for r > R,

kr + S(k)- 1 elk'/, "~ ~C'(k, r) = e ikR /sin I \ k--

~ 2 i k

(2.24)

POLE E X P A N S I O N

d

389

c

~rrO (k, r)l,= g = ikOG(k, R)+ 1,

(2.24a)

q).(r) = q).(R)e ik(r- m.

(2.25)

Further, ~b°(k, 0) = ~o.(0) = 0. Now qJoG and q). satisfy, respectively, the Schr6dinger equations 02

dr 2 ~kCo+ (k z - V)~o° = 0,

(2.26)

d2

dr 2 ~ . + ( k ~ - V ) ~ .

= O.

(2.27)

Multiplying the first equation by q~. and the second by ~k~, subtracting and integrating from 0 to R, we get (k~ - k 2) f:~h~q~.dr + i(k. - k)~C(R)qg.(R) = qg.(R),

(2.28)

c.qg,(r) = res,(ffoG(k, r)) = lim ( ( k - k,)~bCo(k, r)),

(2.29)

now

k--* kn

SO

2k.c.

q~2dr+ ic.q)~(R) =

-

q~.(R),

(2.30)

or, since c. -

-~o.(g)

r

2k.

,

(2.31)

. ~o~(n) ~o.2(r)dr+l

-2k.

-

1.

(2.32)

If instead, we take another residuum, resm(~bo~(k,r)), we obtain

f

: q),(r)~om(r)dr+ i¢p,(R)~o,,(R) k. + k,,

_

(2.33)

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J. BANG

et al.

2.5. C H A R G E D PARTICLES

In the case of charged particles, the solutions for r --* oe do not have the same simple form as for neutrons. We shall still assume that the nuclear potential is 0 for r>R. In the place of the Riccati-Hankel function, we must then introduce the Whittaker functions, W,, so that (2.6) is replaced by

e-~i~l ~k~~c~ = - - - - W_in, t+½(-2ikR)(SlC)(k)W_~n t+½(-2ikr)- W~,,,+~(2ikr)), 2ik "

r > R. (2.34)

Here, t/-

Z1Z2e211 k '

(2.35)

where Zle, Z2e a r e the charges of the projectile and the fixed scatterer, and p their reduced mass. The solutions of the radial equation which continue W_ i~ and 14'+i~ then correspond to the usual f+ and f_, respectively. The natural generalisation of the usual way of finding the singularities of S~(k), is to look at the integral equation, satisfied by f_

f_(k, r) = f~_°)(k, r)+

dpB(l, k, r, p)V(p)f_(k, p),

(2.36)

where the Green function B is given by 1

B(l, k, r, p) = Ttk (fs°)(k' r)ft+°)(k' p ) - ft+°)(k' r)f(-°)(k' p))"

(2.37)

We have here a freedom in the choice of V(p) and f(o). Since the singularity at r = 0 of the point charge Coulomb potential does not occur for realistic charge distributions, the simplest choice is perhaps to say that V is the total potential, nuclear and electrostatic, inside R, and that fro) therefore are combinations of Riccati-Hankel functions, e.g.

fro) = a_(k)h- +a+(k)h +,

r < R.

(2.38) .

f

Note, that in spite of our boundary conditions being different from those of de Alfaro and Regge 19), B is identical to the one of ref. ~9). It is therefore estimated, e.g., by iBl
~'+½(\ i ~ / r

~-t+½ '

b = Im (k).

(2.39)

From (2.36) and (2.38) we must also have

f_(r) = a_ f_ _(r)+a+ f_ +(r),

(2.40)

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391

where .f__(R) = h-(R), etc. so that we can solve the Volterra equation (2.36) for f _ _ and f_ + separately: f _ +(r) = h~:(r)+

dpB(p)V(p)f_ ~:(p).

(2.41)

From this we find, using a restricted V and repeating the argumentation of ref. ~6) or ref. ~9) that f_ ± have no other singularities than the behaviour as (kr) -t for kr ~ 0. The problem of the analytic structure is then reduced to finding a±(k), i.e. to the well-known determination of the proton transmission coefficient 2o). In this problem the centrifugal barrier is of minor importance, and we shall here only look at l = 0. So we have a + e ikR

d-

a_ e - ikR

=

W _ iT ' ½( - -

2ikR),

(2.42) i k ( a + e! kg - a _ e - ikR) =

_ 2 i k W " iT, ½( - 2 i k R ) ,

or, introducing z = - i k R , W = e-Z2z~(1 +iq, 2, 2z), W' = e-Z(½~(iq, O, 2z)-~9(iq, 1, 2z)), a+ = ~ e ~ ( W ~ 2W'), lim ~(iq, c, x/itl)F(i~l + 1 - c ) = 2x ~1-c)/2K 1_c(2x½).

(2.43) (2.44) (2.45)

So, the a have n o ' o t h e r singular point than k = 0, where W has an essential singularity, since it behaves as 1

(2.46)

f ( k ) = f_(k, 0),

(2.47)

f ( k ) = fV(k)/r(iq),

(2.48)

w'

~ (r(in))-

So, the Jost function,

will contain this factor

but fV(k) is holomorphic. Consequently, S can also be written as S(V) = S(C) F ( - i~/) F( + i v ) '

(2.49)

where S v is meromorphic. But ¢~c) is obviously meromorphic, as also the similar expression for the Green function. For those expressions, we can then construct different Mittag-Leffier expressions, with convergence properties practically identical to those of the preceding sections.

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3. Structure problems The resonance functions may be a convenient tool for the calculation of cross sections, e.g. of particle transfer. Another and perhaps more important application is to nuclear structure calculations for such cases where continuum admixtures play an essential role. Such calculations are often conveniently performed in other discrete basis sets, like the Sturmian or oscillator functions. However, when the total energy of the system is above some particle emission threshold these bases are seriously insufficient. We shall therefore study here the use of the resonance functions in nuclear structure problems, looking at two cases, which are each representative of a large class of such problems. In both cases, we shall stick to the simplest possible model of at most one particle being in a continuum state. In this system, in principle, all energy values above a threshold energy are eigenvalues, and we shall in the first case only be concerned with finding the wave function. Let us for simplicity assume that the other degrees of freedom are represented by one particle, which is different from the one with the threshold, so that no antisymmetrization effects occur, and that when the one particle is in the continuum, only one state of the other is - for energy reasons - important. Further, we shall assume zero angular m o m e n t u m for all states involved. So using the Feshbach projection formalism, we write our wave function 0 = t20+P~,

(3.1)

where, in this simple case P~/, = tP2o(r2)

ji~dkc(k)~l,+(k, r,),

(3.2)

QO = ~ cnq),(rl, re).

(3.3)

I1

We shall further assume Hi,e ( = PHP) and HOQ (= QHQ) to be diagonal. The first means, that the 0 +(k) are optical model wave functions, with the important part of the residual interaction (020(r2)lV~es(rlr2)koEo(rz))included in the optical potential V(rl). The second assumption means, that the functions ~o,(r1, rz) are obtained as normalized eigenvectors in a usual bound state shell model calculation. We shall for simplicity assume that the energy scale is such that E = 0 for k = 0, particle 2 in the state q~2o. So we have 2E(k) 6(k( 6 + (k', rl)~a2o(r2)lHee[q+(k, rl)~2o(r2)) -- - -

(nIHQeLm) = E,f,m.

k'),

(3.4) (3.5/

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393

The projection operator P will in our case contain a projection operator

;

,

t

t

(3.6)

P2 = rPEo(r2 ¢PEo(r2)dr2.

We can now from the SchrOdinger equation get the usual set of coupled equations,

(H m , - E)P~k = - HpQQ~b,

(3.7)

(HQQ-- E)QO = - HQpP~b,

(3.8)

(Heo =- PHQ,

Hop =- QHP).

The first equation has the solution

PO = G~ HeqQO +)~p,

(3.9)

where G~- is the Green function for particle 1, corresponding to the optical model Hamiltonian, contained in Hee (in general multiplied by P2), and Zp is a solution of

( H p p - E ) z p = O,

OZp = O.

(3.10)

It therefore obviously must have the form

Xp = c'(k)tP20(r2)~ +(k(E), rl).

(3.11)

The expression for PO is now inserted in the second equation, to obtain

(Hoe - E)QO + HQeG p HeQQ~, + HopXp =

O.

(3.12)

We may now use the expressions (3.3) for QO and (3.11) for )~p, multiply with q~m(r, r2) from the left, and integrate over rx, r 2 to obtain

~c.((E.-E)b.m+ffdr,dr2~o*HeeG~(E)Hpo~o.) + c'(k)f fdr,dr2~o*HQeq~2o(r2)~ ,+ (k(E), r,) = 0

(3.13)

=-- y~ c,((E~ - E(k))cS,m+ Ms~(k)) + c'(k)Nm(k) = O. n

This equation, which is not an eigenvalue equation, since any E(k) in an allowed region gives a solution, determines the c up to a common normalization factor. We see, that the Green function G +(E), and the scattering state ~ +(k) both enter in this expression, respectively in M and N, multiplied with the residual interactions HeQ and HQe and integrated with bound state wave functions. Now, these latter go rapidly to zero with increasing r, and r2, and the residual interactions are of finite range. We can therefore with any desired accuracy replace the infinite integrals in M and N with integrals up to R, where R is of nuclear dimensions.

394

J. B A N G et al.

When this is done, we can introduce the Mittag-Leffler expansions of and ~OC(k):

G~(k) = Z ~°m(r')tP"(rl)) .

(3.14)

'

OC(k, r) = - ~

G+(E(k))

q~"(rl)q~"(R) 2k.(k-k.)

(3.15)

The easiest way to solve the system of equations thus obtained is presumably the following, suggested by Fano 21). The matrix 6.,.(E.-E(k))+Mm.(k ) is diagonalized by an energy dependent transformation, which carries our equation over into

c,(k)(E~(k)- E(k)) + c'(k)N u(k) = O,

(3.16)

c'(k)N,(k) c,(k) - E(k)- Eu(k)

(3.17)

SO

The main part of the calculation is the diagonalisation, which should be performed for all values of k. It is in this respect a great advantage that our Mm, and N,, have the simple forms M,..(k) = ~ 2kz(k_

drldr2r.pm(rlr2)HQpq)20(rz)qh(rl) x ffffdrxdr2q)20(r2)~o,(rl)Hpeq).(rlr2),.

[the ~o.(rl, r2) are of course also of the form

Nm(k) = - ~

(3.18)

Y't~2b71t2q)q(rl)qo12(r2)], drxdr2tp,.HQpq~2o(r2)tpl(rO.

(3.19)

In calculation of the normalization constant it is not so advantageous to use the Mittag-Leffler expansion. We shall here, for our total wave function ~P(E(k),rl, r2), use the same normalization as for ~b+(k, r~), i.e. we shall divide the coefficients with A, calculated from

fofodr,dr2~P*(E(k))~P(E(k'))=½n~(k-k')'A' 2 =

dk"c*(k")Ck,(k") + ~ c.(k)c. (k), 0

n

(3.20)

POLE EXPANSION

395

here q(k") = zc 2

;ofo dr 1

dr20 + *(k", rl)tP*o(r2)P 0

2 f ~ f~drldr2~+.(k,,,rl)CP2o(r2)G+(k)HpQQ~,+c,(k)t~(k_k,,);

/Z

(3.21)

here, we shall use

G+ (k) = ~2f o dk'O +(k"k+2-k'2rl)O+*(k"r'l)

(3.22)

So, the integration over r~ gives

Ck(k") -- k + 2 - - k,,2

drldr2O

(k , rl)HpQq)2o(r2)tPm(rl, r2) + c'(k)cS(k- k"). (3.23)

If, which in general will be the case, HpQ = H~p, 1

~ H Ck(k") -- k+2 _k,,2 E cmN,,(k )+c p(k)6(k-k pr ),

(3.24)

m

and our normalization constant is determined by

x

c'(k')*~(k'-k")+ k'~

k ''~2cmNm(k ) + 2cn(k)c*~(k').

(3.25)

Now

fdk "

1

k+2_k,, 2

N*(k")

1

k_,2_k,, 2 Nm(k")

1 - k , 2 _ kI 2 f (dk" N*(k")k+2_k,,2N,.(k")-N*(k")k_,~ "1_k,,2Nm(k '')) 7g2

+~

6(k- k')N*(k)N,,(k)

4K

1 - k, 2 _ ~-~(M,.n(k)- M*m(k')) ~2

+ ~-~ (~(k- k')N*(k)N,,,(k).

(3.25a)

396

J. B A N G et al.

If we now multiply (3.13) by c,,(k ), sum over m and subtract the complex conjugate sum with k and k' exchanged, we get

0 = ~ c,.(k )c.(k)((E(k ) - E(k))6m. + Mm.(k ) - M.,.(k )) m , rl

+c'(k,) ~ c*(k)N.,(k)- c'*(k) ~ c,.(k')N*(k'), Wit

(3.13a)

m

or

1

c*(k')c,(k) - k2_ k, 2 n

x ~ (c'(k')c*(k)Nm(k) - c'*(k)cm(k')N*(k' ) + ~ c*(k')cn(k)(Mm.(k ) - M,*m(k'))), ra

(3.13b)

n

and inserting this in (3.25) we get IA]2 = _2 n IC,] 2 -}- ~/I~ 1~

cm(k)N,,(k)l 2.

(3.26)

m

In a concrete calculation, it will be natural to chose A = 1 or c' = 1. The qualitative picture, obtained in this way, has been discussed many times. The cross section of for example elastic scattering, will, due to the residual interactions, have a large number of narrow resonances. Averaging over an energy interval which is larger than these widths, one regains a cross section corresponding to the ~b+, however with somewhat larger single particle resonance widths. A different situation is met, if the residual interaction has matrix elements mainly to such states, where the other degrees of freedom - in the above model represented by the particle 2 - takes up a small amount of the energy only. This will often be the case with collective degrees of freedom. In such cases, the assumption of only one continuum must be given up, but on the other hand, the admixture of discrete states will be large only, when these states, as well as the total energy, lies near the particle emission threshold. As an extreme but not unrealistic example, one can think of the coupling of the single particle states to a static deformed field, as treated in ref. 22). In such cases, the main effect of the residual interaction can be described as the shift of positions and widths of resonances, r A method to calculate these shifts in perturbation theory was given in ref. 23). The starting point is here the Dyson relation between G~- and G ÷, pertaining to the unperturbed and the perturbed potential respectively: t G ÷ = G g + G 0+ H G

+

,

(3.27)

where H' is the perturbation. Assuming again that the perturbation is active inside R only, we can again introduce Mittag-LetIler expansions for both G ÷ and G~-.

POLE

EXPANSION

397

Equating the residues at k = k,, one gets ,

(,On =

o

o ,

~°mH q~ndr"

(3.28)

qg. = Z )-i~°~,~',

(3.29)

o

2k,.(k.-k,.)

If we put

/-/' = 2U, k, = Z 2'K(,i', i=0

i=0

and equate the coefficients of each power of 2, one gets a perturbation series for wave functions and energies. It should be noted, however, that there is no term-by-term correspondence between this series, and the usual perturbation series, even when we look at the contributions from bound state residua only. This is due to the nonorthogonality of the resonance functions. Only the first order corrections in the energy are in agreement. If an exact diagonaiisation in a truncated space is preferred, a similar problem is met. The equation N

(H- E)~ t

....

=

(H 0 + H ' - E) ~ c,q~, = 0

(3.30)

n

leads, e.g. to a matrix eigenvalue equation, when multiplied by N independent functions and integrated over all angles and over r from 0 to R. Due to the non-orthogonality, E will appear with non-diagonal terms as well as diagonal. Between two bound states, its non-diagonal contributions will be small, so that such eigenvectors which consist mainly of bound state components must resemble the usual ones. In general the non-diagonal terms are, however, large. The non-orthogonality is not merely a calculational inconveniency, but also leads to a freedom in the choice of the set of equations, satisfied by the eigenvectors. A natural choice seems to be to assume that a certain basis n = n 1 ... n u is sufficient for the expansion of ~O, and that the same basis functions are used for the matrix (nllHIn2). It should be noted that states with different /-values are trivially orthogonal, and that therefore in the approximation of one pole for each /-value in the expansion, no problem of the type mentioned arises 22). The ambiguity in the choice of a truncated problem is connected with the wellknown absence of minimum principles for scattering states. However, stationarity principles do exist, and a better starting point for the truncation is obtained when we start from a stationary expression, and make a truncation of that expression. Now, the expression (3.27) above can just be obtained from the variation of an expression for G+_G~

= G o+H G, +,

6(GgH'G++G+H'Go-G+H'G++G+H'G~H'G

where the variation is in the parameters of G +.

+) = 0,

(3.31)

J. BANG et al.

398

Since the expression is stationary, a small change in these parameters, like a truncation with sufficiently many terms, must lead to an expression which is near to the true one. Making now our variation with such a truncated expression, we obtain the equation

c,qg, = n

2k,,(k- kin) ~°mn'( c,q~,)dz,

(3.32)

m

or, defining ~,, = tp~/x/km, ~,, = x~kmc,,, ,.

2(k-k,.)

~b,.H'(

e,~b.)dr.

(3.33)

Equating the coefficients of ~,, we get

(k-k,,)?,. = ~ c,(q~, IH lop,,),

(3.34)

which is obviously an eigenvalue problem of the usual type, though non-Hermitian (but symmetric). 4. Numerical results As an illustration of the general results obtained in sect. 2 we shall in this paragraph look at the potential Voh(r- R), which was considered in detail in refs. 23, 26), and at a rectangular well. It should be noticed that numerical investigations of the convergence of the Mittag-Leffler expansion of the S-matrix are nearly absent in the literature [see, though, the article by WeidenmiJller 24)], and for the wave functions, such investigations were never carried out. We give here some formulae pertaining to the above-mentioned potentials, and limiting ourselves to the S-states. 4.1. THE h-POTENTIAL The exact expression for the S-matrix is

S(k) = 1 + Vo(1- e- 2ikR)/2ik 1 + Vo(e2ikR - 1)/2ik

(4.1)

For the resonance wave function we have for r < R,

(2(Vo_2ik~) q~(r) =

~

1 + R ( ~ o£ - 2/k~)/ sin (kar),

(4.2)

with k~ determined by 1 + Vo(e2ik~g- 1)/2ik~ = 0.

(4.2a)

'POLE EXPANSION

399

T h e expansion of the S-matrix is

[ k" ~ (k'] "+1 r~ ] S((P))(k) = e-2'kR S"(O)+kS'R(O)+ "" + p-i S~)(0)+~1= \-k-aJ k~-k~ 2k~ r~ = iVo[ 1 + R(Vo _ 2ika)] '

2iR S~(O)- - I+VoR'

S~(O) = 1, S~(O) = --

(4.3)

4R 2

4iR3(2 - VoR )

SHI(o) --

(1 "l- FoR) 2 '

(1 -'1- VoR) 3

sIV(o)

=

16R4(1 -- 2 VoR ) (1 + VoR) 4

(4.3a)

T h e wave function of the continuous spectrum is e ikR sin (kr) 0G(k, r) = k(1 + Vo(e 2'kR - 1)/2ik) '

r <= R,

(4.4}

SO

r ~k(°)(O, r) - , , ~ , + 1 0

~b(1)(O,r) -

irR (1 + VoR )2 '

(4.5)

and we have the expansions ~bl(k, t)

~k"(k, r) -

r

l + VoR

V~ k (pa(R)q)~(r)

1

r ikrR 1 + Vo~ + (1 + VoR) 2

k~ k~(k-k2) ~ k ½a~==, k ~

2 (p~(R)tp2(r) ka(k-ka) '

so we can c o m p a r e the exact expressions for the S-matrix nad the wave functions with the corresponding Mittag-Leffler expansions. In table 1 the values of k. are given for Vo = 40 and - 40 f m - 1 and in table ]2 the values of the exact S-matrix and the S-matrix o b t a i n e d from the Mittag-Leffler expansion for p = 1, 3 with Inl =< 7 (Vo = 40 fm-1). Fig. 1 shows S . . . . t sappr for p , = 1, 2, 3 and Int < 7 (dotted curve) and Inl < 1 (full drawn). F r o m the tables and figures it is seen, that the convergence is strongly' dependent on p and becomes faster for larger p-values. Tables 3a, b and c show the exact wave function and the Mittag-LetIler a p p r o x i m a tion for the wave function with p = 0 , 1 and I n l < 1, Inl < 7 and Inl =<22, for V = 40 f m - 1 . In tables 4a, b and c the corresponding results for V = - 4 0 f m - ~ are shown. F r o m these tables, it is seen that for Int < 22 the Mittag-Leffier expansion expression for the wave function is correct with four digits for k between 0 and 5 f m - 1. A similar g o o d convergence is seen for the G r e e n function.

400

J. B A N G

et al.

TABLE 1 P o s i t i o n o f poles, V = Vo6(r- R) V0 = 4 0 f m

I 2 3 4 5 6 7 8 9 I0 I1 12 13 14 15 16 17 18 19 20 21 22

look

expression

Vo = - 4 0 f m

I,R = 1

Re k,

Im k ,

Re k,

lm k,

0 . 3 0 6 6 E + 01 0 . 6 1 3 4 E + 01 0 . 9 2 0 9 E + 01 0.1229E +02 0.1538E+02 0.1847E+02 0.2158E+02 0 . 2 4 6 8 E + 02 0 . 2 7 8 0 E + 02 0.3091 E + 02 0.3403E+02 0.3716E+02 0 . 4 0 2 8 E + 02 0.4341E+02 0 . 4 6 5 4 E + 02 0 . 4 9 6 7 E + 02 0 . 5 2 8 0 E + 02 0 . 5 5 9 3 E + 02 0.5906E+02 0 . 6 2 2 0 E + 02 0 . 6 5 3 3 E + 02 0 . 6 8 4 7 E + 02

- 0 . 5 6 6 7 E - 02 - 0 . 2 1 9 7 E - 01 - 0 . 4 7 0 9 E - 01 - 0 . 7 8 6 6 E 01 -0.1143E+00 -0.1522E+00 -0.1908E+00 - 0 . 2 2 9 1 E + 00 - 0 . 2 6 6 6 E + 00 - 0 . 3 0 2 9 E + 00 -0.3378E+00 -0.3712E +00 - 0 . 4 0 3 2 E + 00 - 0.4337E +00 - 0 . 4 6 2 8 E + 00 - 0 . 4 9 0 7 E + 00 - 0 . 5 1 7 3 E + 00 - 0 . 5 4 2 8 E + 00 -0.5671E+00 - 0 . 5 9 0 5 E + 00 - 0 . 6 1 3 0 E + 00 - 0 . 6 3 4 6 E + 00

0 0 . 3 2 2 1 E + 01 0 . 6 4 3 9 E + 01 0 . 9 6 4 9 E + 01 0.1285E+02 0.1604E+02 0.1923E+02 0 . 2 2 4 1 E + 02 0 . 2 5 5 8 E + 02 0 . 2 8 7 5 E + 02 0.3192E+02 0.3808E+02 0 . 3 8 2 4 E + 02 0.4140E+02 0 . 4 4 5 5 E + 02 0 . 4 7 7 1 E + 02 0 . 5 0 8 6 E + 02 0.5401 E + 02 0.5716E+02 0.6031 E + 02 0 . 6 3 4 6 E + 02 0 . 6 6 6 1 E + 02

0 . 2 0 0 0 E + 02 - 0 . 6 5 6 3 E - 02 - 0 . 2 5 2 3 E - 02 - 0.5340E-01 - 0 . 8 7 9 6 E - 01 -0.1261E+00 -0.1658E+00 - 0 . 2 0 5 6 E + 00 0 . 2 4 4 7 E + 00 - 0.2825 E + 00 -0.3188E +00 -0.3535E +00 - 0 . 3 8 6 6 E + 00 -0.4182E+00 - 0 . 4 4 8 3 E + 00 - 0 . 4 7 6 9 E + 00 - 0.5043 E + 00 - 0 . 5 3 0 5 E + 00 -0.5555E +00 - 0 . 5 7 9 4 E + 00 - 0 . 6 0 2 4 E + 00 - 0 . 6 2 4 4 E + 00

4.2. R E C T A N G U L A R We

~,R = 1

WELL

V(r) = - V o f o r r < R a n d

at the potential for the S-matrix

and

S(k) =

wave

function

V(r) = 0 , f o r r > R . T h e e x a c t

(for r <

x c o s x R + ik s i n x R

e-

R) are

2ikR,

(4.6)

x c o s x R - ik s i n x R sin xr ~C(k, r) =

where

k2=

function

(2m/h2)E,

we have

xg = (2m/h2)Vo

for r <

and

x2=

ka determined

kZ+x~.

For

the

resonance

wave

R,

s i n x~r,

~oz(r) = N / ~

with

(4.7)

x c o s x R - ik s i n x R '

(4.8)

by ctg ~

R

=

ik,v

(4.8a)

POLE EXPANSION

401

TABLE 2a The S-matrix expansion for different k, retaining only the first pole Exact

1.00 3.00 5.00 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

p = 1

p = 3

Re S

Im S

Re S

Im S

Re S

Im S

-0.3709 0.8270 -0.9502 0.2214 -0.9562 0.7291 0.9570 0.9909 0.9980 0.9997 1.0000 1.0000 1.0000 1.0000 0.9999

-0.9287 0.5623 0.3118 0.9752 -0.2926 -0.6844 - 0.2900 -0.1345 -0.0625 -0.0262 -0.0083 -0.0010 -0.0004 -0.0042 -0.0111

-0.3715 0.8378 -0.9628 0.2325 -0.9451 0.7404 0.9682 1.0022 1.0093 1.0109 1.0112 1.0113 1.0113 1.0112 1.0112

-0.9296 0.5616 0.3563 0.9731 -0.2949 -0.6869 -0.2929 -0.1376 -0.0658 -0.0298 -0.0122 -0.0052 -0.0048 -0.0089 - 0.0161

-0.3709 0.8277 -0.9480 0.2221 -0.9555 0.7299 0.9577 0.9916 0.9988 1.0004 1.0007 1.0007 1.0007 1.0007 1.0006

-0.9287 0.5618 0.3298 0.9746 -0.2932 -0.6850 -0.2907 -0.1352 -0.0631 -0.0269 -0.0091 -0.0018 -0.0012 -0.0050 - 0.0120

TABLE 2b The S-matrix expansion for different k, with the first seven poles in the expansion Exact

1.00 3.00 5.00 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

p = 1

p = 3

Re S

Im S

Re S

Im S

Re S

Im S

-0.3709 0.8270 -0.9502 0.2214 0.9562 -0.7291 0.9570 0.9909 0.9980 0.9997 1.0000 1.0000 1.0000 1.0000 0.9999

-0.9287 0.5623 0.3118 0.9752 -0.2926 -0.6844 -0.2900 -0.1345 -0.0625 -0.0262 - 0.0083 - 0.0010 - 0.0004 - 0.0042 -0.0111

-0.3712 0.8319 -0.9619 0.2267 -0.9509 0.7345 0.9625 0.9964 1.0036 1.0052 1.0056 1.0056 1.0057 1.0057 1.0057

-0.9292 0.5636 0.3203 0.9759 -0.2919 -0.6838 -0.2896 -0.1342 -0.0622 -0.0261 - 0.0083 - 0.0011 - 0.0006 - 0.0045 -0.0116

-0.3709 0.8270 -0.9503 0.2214 -0.9562 0.7292 0.9570 0.9909 0.9981 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000

-0.9287 0.5623 0.3119 0.9752 -0.2926 -0.6844 -0.2900 -0.1345 -0.0625 -0.0262 - 0.0083 - 0.0010 - 0.0004 - 0.0042 -0.011 !

402

J. BANG et al. ~0-I

I sexOcLsP S exact I

p=l p=2

p=3

-2

p=l

I0

t6 3 /.

/

/

/

i,

p'-2

/ / /

p=3

/ /

1¢

//

,/J

/

/

/

/

/

/ / /

16

0

1

2.

3

e,

5

Fig. 1. The value of I(S~Xa¢'-SP)/SeXa"J as a function of k for different Mittag-Leffier expansions, for V(r) = Vob(r- R), V o =

40 fm- l, R = 1.

The Mittag-Leffler expansion of the S-matrix is

SP(k)

= e - 2ikR

SR(O)~-kSrR(O)"+-

" " " ~- p i

S~)(O)

-i

=1 k k2J

k-k~'

(4.9)

SR(O) = 1, 2i

SI(0) = - - tg xoR, Xo

S~(0) = 6i

4 x~6 tg2 x°R'

S~'(O) = Xo-Z~(xoR - tg xoR)(1 + tg 2 xoR ) -

6i

()x~tg 3 xoR ,

(4.9a)

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

3.0

5.0

00 00 00 00 01

- 0 . 2 2 4 0 E - 01 -0.2421E-01 - 0 . 3 7 5 7 E - 02 0 . 2 0 1 5 E - 01 0 . 2 5 5 3 E - 01

0.2059E + 0.3399E + 0.3552E + 0.2464E + 0.5147E-

0 . 5 8 0 6 E - 02 0.1138E-01 0 . 1 6 5 0 E - 01 0 . 2 0 9 6 E - 01 0 . 2 4 5 9 E - 01

H e r e E + 02 ~- 102, etc.

0.2 0.4 0.6 0.8 1.0

.0

R e ~b

Exact

01 01 01 01 02

03 03 03 03 03

- 0.2908 E - 02 -0.3142E-02 - 0 . 4 8 7 7 E - 03 0 . 2 6 1 5 E - 02 0 . 3 3 1 4 E - 02

0.3259E0.5380E0.5622E0.3899E 0.8146E-

0.1428E 0.2800E0.4060E 0.5158E0.6050E-

I m ~k

00 00 00 00 01

02 01 0i 01 01

- 0 . 9 6 8 2 E - 02 -0.1406E-01 - 0 . 9 7 5 8 E - 02 0 . 3 4 3 0 E - 02 0 . 2 2 4 8 E - 01

0.2078E + 0.3416E + 0.3545E + 0.2435E + 0.5086E-

0.5960E 0.1153E0. i 645 E 0.2071 E 0.2453E-

R e ff

p = 0

01 01 01 01 02

- 0 . 6 1 3 2 E - 03 -0.1015E-02 - 0 . 1 0 6 6 E - 02 - 0 . 7 4 7 8 E - 03 - 0 . 1 7 0 5 E - 03

0.3304E0.5432E0.5625E0.3815E 0.6444E-

0 . 2 4 9 0 E - 03 0.4116E-03 0 . 4 3 1 3 E - 03 0 . 3 0 1 2 E - 03 0 . 6 6 1 5 E - 04

Im ~

00 00 00 00 01 - 0 . 9 6 8 2 E - 02 -0.1406E-01 - 0 . 9 7 5 8 E - 02 0 . 3 4 3 0 E - 02 0 . 2 2 4 8 E - 01

0.2078E + 0.3416E + 0.3545E + 0.2435E + 0.5086E-

0 . 5 9 6 0 E - 02 0.1153E-01 0 . 1 6 4 5 E - 01 0.2071 E - 01 0 . 2 4 5 3 E - 01

R e ¢,

p = 1

W a v e f u n c t i o n s e x p a n d e d f o r r _-< R a n d d i f f e r e n t k - v a l u e s r e t a i n i n g o n l y the first p o l e in t h e e x p a n s i o n ( V 0 = 40)

TABLE 3a

01 01 01 01 02

03 03 03 03 03

- 0. I 124E - 02 -0.1653E-02 - 0 . 1 1 9 7 E - 02 0 . 2 9 3 1 E - 03 0 . 2 5 0 9 E - 02

0.3274E0.5394E0.5617E0.3877E 0.8052E-

0.1468E 0.2839E0.4050E 0.5094E0.6021 E -

Im

7~

), 7t

rn rn

t"

O

0.2059E + 00 0.3399E + 00 0.3552E + 00 0.2464E + 00 0.5147E-01

- 0.2240E - 01 -0.2421E-01 -0.3757E-02 0 . 2 0 1 5 E - 01 0.2553E - 01

0.2 0.4 0.6 0.8

1.0

0.2 0.4 0.6 0.8 1.0

3.0

02 01 01 01 01

0.5806E0.1138E 0.1650E 0.2096E 0.2459E -

0.2 0.4 0.6 0.8 1.0-

R e qJ

1.0

H e r e E + 0 2 ~- 102, etc.

5.0

TABLE 3b

Exact

01 01 01 01 02

03 03 03 03 03

- 0.2908E - 02 -0.3142E-02 -0.3877E-03 0 . 2 6 1 5 E - 02 0 . 3 3 1 4 E - 02

0.3259E0.5380E0.5622E0.3899E0.8146E-

0.1428E0.2800E 0.4060E 0.5158E 0.6050E-

I m ~b

00 00 00 00 00

02 01 01 01 01

- 0.2244E - 01 -0.2421E-01 - 0 . 3 6 9 7 E - 02 0 . 2 0 1 7 E - 01 0.2545E - 01

0.2059E + 0.3399E + 0.3552E + 0.2464E + 0.5144E-

0.5804E0.1138E 0.1650E 0.2096E 0.2459E-

R e qJ

p = 0

01 01 01 01 02

03 03 03 03 03

- 0.2994E - 02 -0.3115E -02 -0.3575E-03 0 . 2 5 1 5 E - 02 0 . 1 8 6 9 E - 02

0.3254E0.5382E0.5629E0.3893E0.7287E-

0.1264E0.2853E 0.4308E 0.4961E 0.3198E-

Im ~

02 01 01 01 01

- 0.2244E - 01 -0.2421E-01 - 0 . 3 6 9 7 E - 02 0 . 2 0 1 7 E - 01 0 . 2 5 4 5 E - 01

0.2059E + 00 0.3399E + 00 0.3552E + 00 0.2464E + 00 0.5144E-01

0.5804E0.1138E0.1650E 0.2096E 0.2459E-

Re qJ

p=l

W a v e functions e x p a n d e d for r < R a n d d i f f e r e n t k-values with the first seven poles in the e x p a n s i o n ( V o = 40)

03 03 03 03 03

- 0.2912E - 02 -0.3141E-02 -0.4815E-03 0 . 2 6 1 4 E - 02 0 . 3 2 9 4 E - 02

0 . 3 2 5 9 E - 01 0.5380E - 01 0 . 5 6 2 2 E - 01 0 . 3 8 9 9 E - 01 0.8142E - 0 2

0.1428E0.2800E 0.4060E 0.5158E 0.6049E -

l m ~p

Z

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

1.0

3.0

5.0

- 0.2240E - 01 -0,2421E-01 -0.3757E-02 0.2015E-01 0.2553E-01

0.3399E+00 0.3552E+00 0.2464E + 00 0.5147E-01

0.2059E + 00

0 . 5 8 0 6 E - 02 0.1138E-01 0.1650E-01 0.2096E - 01 0.2459E-01

Re ¢

Exact

-0.3142E-02 - 0.4877E - 03 0 . 2 6 1 5 E - 02 0.3314E-02

- 0.2908E - 02

0.3259E-01 0.5380E-01 0.5622E-01 0.3899E-01 0.8146E-02

0 . 1 4 2 8 E - 03 0.2800E - 03 0.4060E - 03 0.5158E-03 0.6050E - 03

Im ¢,

- 0.2240E - 01 -0.2421E-01 - 0 . 3 7 5 6 E - 02 0.2015E-01 0.2553E-01

0.2059E + 00 0.3399E + 00 0.3552E+00 0.2464E + 00 0.5147E-01

0.5806E - 02 0.1138E-01 0 . 1 6 5 0 E - 01 0.2096E - 01 0.2459E - 01

Re

p=O

-0.2916E-02 -0.3148E-02 -0.4758E-03 0.2643E-02 0.2758E - 02

0.3259E-01 0 . 5 3 8 0 E - 01 0.5622E-01 0.3901 E - 01 0.7813E-02

0.1412E-03 0.2789E - 03 0.4084E - 03 0.5213E-03 0.4941E - 03

Im

W a v e f u n c t i o n with Inl ~ 22 ( V o = 40)

TABLE 3C

- 0 . 2 2 4 0 E - 01 -0.2421E-01 - 0.3756E - 02 0.2015E-01 0.2553E-01

0.2059E + 00 0.3399E + 00 0.3552E+00 0.2464E + 00 0.5147E-01

0.5806E - 02 0.1138E-01 0.1650E-01 0 . 2 0 9 6 E - 01 0.2459E-01

R e ~b

p=l

- 0.2908 E - 02 -0.3142E-01 - 0.4876E - 03 0.2615E-02 0.3313E-02

0.3259E-01 0.5380E-01 0.5622E-01 0.3899E-01 0.8146E-02

0.1428E - 03 0.2800E - 03 0.4060E - 03 0.5158E-03 0.6050E - 03

Im

©

4~

Z

6

(/1

Z

"0

X

rn

t"

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

5.00

1.0

0.2 0.4 0.6 0.8

3.0

1.0

0 . 2 0 8 5 E - 01 0 . 2 2 5 3 E - 01 0 . 3 4 9 7 E - 02 -0.1871E-01 - 0 . 2 3 7 6 E - 01

-0.6539E-01 -0.1079E +00 - 0 . 1 1 2 8 E + 00 - 0 . 7 8 2 2 E - 01 -0.1634E-01

-0.5995E-02 - 0.1175 E - 01 -0.1704E-01 - 0.2165E - 01 - 0 . 2 5 3 9 E - 01

Re ~b

Exact

- 0 . 2 5 1 4 E - 02 - 0 . 2 7 1 6 E - 02 - 0 . 4 2 1 6 E - 03 0.2261E-02 0 . 2 8 6 4 E - 02

0.3213E-02 0.5304E-02 0 . 5 5 4 2 E - 02 0 . 3 8 4 4 E - 02 0.8031E-03

-0.1523E-03 0.2986 E - 03 0.4329E-03 0.5500E - 03 0.6451 E - 03

lm ~

TABLE 4a

0 . 2 0 5 4 E - 01 0 . 2 3 0 4 E - 01 0.2981 E - 02 -0.1842E-01 - 0 . 2 4 1 6 E - 01

-0.5128E-02 -0.1026E-01 -0.1538E-01 - 0 . 2 0 5 0 E - 01 -0.2509E-01

-0.5128E-02 - 0.1026 E - 01 -0.1538E-01 - 0.2051E - 01 - 0 . 2 5 5 8 E - 01

Re ~

p=O

- 0 . 2 3 1 3 E - 02 - 0 . 3 0 5 7 E - 02 - 0 . 5 2 6 5 E - 04 0.1994E-02 0 . 5 0 5 5 E - 02

0.4126E-09 0.2253E-07 0.1230E - 05 0 . 6 7 1 7 E - 04 0.3667E-02

0.1403E-09 0.7661 E - 08 0.4183E-06 0.2284E - 04 0 . 1 2 4 7 E - 02

Im ¢,

W a v e f u n c t i o n w i t h Inl = 1 ( V o = - 4 0 )

0 . 2 0 5 4 E - 01 0 . 2 3 0 4 E - 01 0.2981 E - 02 -0.1842E-01 - 0 . 2 4 1 6 E - 01

-0.1026E-01 - 0 . 1 5 3 8 E - 01 - 0 . 2 0 5 0 E - 01 - 0 . 2 5 0 9 E - 01

-0.5128E-02

-0.5128E-02 - 0.1026 E - 01 -0.1538E-01 - 0.2051E - 01 - 0 . 2 5 5 8 E - 01

Re ¢,

p=l

- 0 . 2 4 8 3 E - 02 - 0.2771 E - 02 - 0 . 3 5 6 9 E - 03 0.2196E-02 0 . 2 9 6 0 E - 02

0.3945E-03 0.7890E-03 0 . 1 1 8 3 E - 02 0 . 1 5 7 6 E - 02 0.1890E-02

0.1315E-03 0.2630E - 03 0.3945E-03 0.5259E - 03 0 . 6 5 4 3 E - 03

Im

> Z 0

-0.599 E-02 -0.1175E-01 -0.1704E-01 -0.2165E-01 - 0.2539E-01

-0.6539E-01 - 0.1079E+00 -0.1128E+00 - 0.7822E-01 -0.1634E-01

0.2085 E - 01 0 . 2 2 5 3 E - 01 0 . 3 4 9 7 E - 02 -0.1875E-01 - 0 . 2 3 7 6 E - 01

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

1.0

3.0

5.0

R e ~,

Exact

- 0.2514E - 0.2716E- 0.4216E 0.2261E-02 0.2864E02

02 02 03

0.3213E-02 0.5304E-02 0.5542E-02 0.3844E-02 0.8031E-03

0.1523E-03 0.2986E-03 0.4329E-03 0.5500E-03 0.6451E-03

I m ~,

0.2080E 0.2259E0.4514E -0.1889E-0l - 0.2387E01

01 01 02

- 0 . 6 5 4 0 E - 01 -0.1079E+00 -0.1128E+00 -0.7827E-01 -0.1638E-01

-0.5997E-02 -0.t 175E-01 -0.1704E-01 -0.2165E-01 -0.2540E-01

R e ~k

p=0

- 0 . 2 4 1 9 E - 02 - 0 . 2 8 0 6 E - 02 - 0.4843 E - 03 0.2588E-02 0 . 4 4 0 2 E - 02

0.3268E-02 0.5252E-02 0.5504E-02 0.4035E-02 0.1716E-02

0.1702E-03 0.2814E-03 0.4212E-03 0.6128E-03 0.9479E-03

I m ~b

W a v e f u n c t i o n w i t h [nl < 7 ( V o = - 4 0 )

TABLE 4 b

0 . 2 0 8 0 E - 01 0 . 2 2 5 9 E - 01 0 . 3 5 1 4 E - 02 -0.1889E-01 - 0 . 2 3 8 7 E - 01

-0.6540E-01 -0.1079E+00 -0.1128E+00 -0.7827E-01 -0.1638E-01

- 0 . 5 9 9 7 E - 02 -0.1175E-01 -0.1704E-01 -0.2165E-01 - 0 . 2 5 4 0 E - 01

Re ~

p=l

- 0.2509E - 0.2720E- 0.4262E 0.2274E0.2889E -

02 02 03 02 02

0.3214E-02 0.5303E-02 0.5541E-02 0.3847E-02 0.8082E-03

0 . 1 5 2 4 E - 03 0.2985E-03 0.4329E-03 0.5501E-03 0.6453E-03

I m ~k

4~

Z

:> Z

© r" m

-0.6539E-01 - 0.1079E + 00 - 0.1128E + 00 - 0 . 7 8 2 2 E - 01 -0.1634E-01

0 . 2 0 8 5 E - 01 0.2253E-01 0.3497E - 02 -0.1875E-01 - 0 . 2 3 7 6 E - 01

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

3.0

5.0

1.0

- 0 . 5 9 9 5 E - 02 -0.1175E-01 -0.1704E-01 - 0 . 2 1 6 5 E - 01 - 0.2539E01

1.0

0.2 0.4 0.6 0.8

R e ff

Exact

- 0 . 2 5 1 4 E - 02 -0.2716E-02 - 0 . 4 2 1 6 E - 03 0.2261E-02 0 . 2 8 6 4 E - 02

0.3213E-02 0.5304E - 02 0.5542E - 02 0.3844E- 02 0.8031E-03

0 . 1 5 2 3 E - 03 0 . 2 9 8 6 E - 03 0.4329E-03 0 . 5 5 0 0 E - 03 0.6451 E - 03

I m ~k

Wave

TABLE 4C

01 01

02 01

0 . 2 0 8 5 E - 01 0.2253E-01 0.3497E - 02 -0.1875E-01 - 0 . 2 3 7 7 E - 01

-0.6539E-01 - 0.1079E + 00 - 0.1128E + 00 - 0 . 7 8 2 2 E - 01 -0.1634E-01

- 0.5995 E -0.1175E-0.1704E-01 - 0.2165E- 0.2539E-

R e qJ

p=0 I m g,

-40)

- 0 . 2 5 0 5 E - 02 -0.2710E-02 - 0 . 4 3 1 3 E - 03 0.2226E -02 0 . 3 4 3 2 E - 02

0.3218E-02 0.5308E - 02 0.5536E - 02 0.3823E- 02 0.1143E-02

0 . 1 5 4 0 E - 03 0 . 2 9 9 9 E - 03 0.4310E-03 0 . 5 4 3 0 E - 03 0 . 7 5 8 4 E - 03

f u n c t i o n w i t h Inl < 22 ( V o =

01 01

02 01

0 . 2 0 8 5 E - - 01 0.2253E-01 0 . 3 4 9 7 E - 02 -0.1875E-01 - 0 . 2 3 7 7 E - 01

-0.6539E-01 - 0 . 1 0 7 9 E + 00 - 0 . 1 1 2 8 E + 00 - 0 . 7 8 2 2 E - 01 -0.1634E-01

- 0.5995E -0.1175E-0.1704E-01 - 0.2165E- 0.2539E-

Re ¢

p=l

03 03 03 03 03

-- 0 . 2 5 1 4 E - 02 -0.2716E-02 - 0 . 4 2 1 6 E - 03 0.2261E-02 0 . 2 8 6 5 E - 02

0.3213E-02 0 . 5 3 0 4 E - 02 0 . 5 5 4 2 E - 02 0 . 3 8 4 4 E - 02 0 . 8 0 3 3 E - 03

0.1523E 0.2986E0.4329E0.5500E0.6452E-

lm

> Z

POLE EXPANSION

409

and for the wave function we have the expansion

~6(k, r)" = ~b~(O, r) + k~b6(O, r)' + "'"

+ -p-i t~6(O' r)(")- 1 ~ 2 ~=1 \ k J

1 tpz(R)tpz(r) k~(k-k~) ' (4.10)

ff~(0, r) -

~,G(O,r),_

1 sin Xor x o cos x o R '

rcosxor

x 2 cos Xor

1

~bC(O, r) t _

i sin Xor sin x ° R ,

x2

sinxor

x 3 cos 3 xoR

(4.10a)

cos 2 xoR

[l_xoRsinxoRcosxoR+sinZxoR].

There is an interesting relation between fie(k, R F and S p+I"

S p+ l(k)e2ikR -- 1 2ik

~k6(k' R)P =

(4.11)

In table 5 the values of k~ are given for Vo = 48.72 MeV, flo = 4.6 and flo = 5.0 (where flo = xoR). In table 7 the values of the exact S-matrix and the Mittag-Leffler expansion of it are given for p = 2, 3 with n < 1, 4, 7 and 22 (flo = 4.6). Fig. 2 shows TABLE 5 Positions o f the poles for the r e c t a n g u l a r potential wells flo = 4.6

flo = 5.0

N

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

R e k~

I m k~

Re k~

I m k~

0.5703E - 01 0 0.2066E+01 0.3281E+01 0.4413E + 01 0.5513E+01 0.6597E + 01 0.7670E + 01 0.8738E + 01 0.9801E + 01 0.1086E + 02 0.1192E + 02 0.1298E+02 0.1403E + 02 0.1509E + 02 0.1614E+02 0.1719E+02 0.1824E+02 0.1929E+02 0.2034E + 02 0.2140E + 02 0.2245E + 02

- 0.3335E + 00 0.1275E + 01 - 0.4623E + 0 0 - 0.5563E + 0 0 - 0.6303E + 00 -0.6911E+00 - 0.7428E + 00 - 0.7878E + 00 - 0.8274E + 00 - 0.8630E + 00 - 0.8951E + 00 - 0.9245E + 00 -0.9515E+00 - 0.9765E + 00 - 0.9998E + 00 -0.1021E+01 -0.1042E+01 -0.1061E+01 -0.1080E+01 - 0.1097E + 01 -0.1113E+01 - 0.1129E + 01

0 0 0 0.1808E+01 0.2961 E + 01 0.4018E+01 0.5038 E + 01 0.6040E + 01 0.7032E + 01 0.8017E + 01 0.8998E + 01 0.9975E + 01 0.1095E+02 0.1192E + 02 0.1290E + 02 0.1387E+02 0.1484E+02 0.1580E+02 0.1677E+02 0.1774E + 02 0.1871E+02 0.1968E + 02

0.2955E + 00 - 0.8756E + 00 0.131 I E + 0 1 -0.4051E +00 - 0.4900E + 00 -0.5570E+00 - 0.6124E + 00 - 0.6596E + 00 - 0.7006E + 00 - 0.7368E + 00 - 0.7694E + 00 - 0.7988 E + 00 -0.8257E+00 - 0.8505E + 00 - 0.8734E + 00 - 0 . 8 9 4 8 E + 00 -0.9148E+00 -0.9336E+00 -0.9513E+00 - 0.9680E + 00 - 0.9839E + 0 0 - 0.9990E + 00

r

0.60 1.20 1.80 2.40 3.00

0.60 1.20 1.80 2.40 3.00

0.60 1.20 1.80 2.40 3.00

k

1.0

3.0

5.0

- 0 . 7 0 6 5 E - 03 0.1413E-02 -0.2119E-02 0 . 2 8 2 5 E - 02 -0.3532E-02

- 0 . 2 2 5 9 E + 00 0.1968E+00 0 . 5 4 4 6 E - 01 - 0 . 2 4 4 3 E + 00 0.1583E+00

0 . 5 3 0 1 E + 00 0 . 4 8 2 4 E + 00 -0.9101E+01 - 0 . 5 6 5 2 E + 00 -0.4234E +00

Re •

Exact

0.1248E-04 -0.2496E-04 0.3744E-04 - 0 . 4 9 9 2 E - 04 0.6240E-04

-0.1636E+00 0.1425E+00 0 . 3 9 4 3 E - 01 -0.1768E+00 0.1146E +00

- 0 . 2 9 3 0 E + 00 - 0 . 2 6 6 7 E + 00 0.5031E-01 0.3125E+00 0.2341E+00

I m ~,

0.1487E+00 0.3275E-01 -0.3036E +00 - 0 . 5 0 9 3 E + 00 - 0 . 2 6 0 4 E + 00

0.1807E+00 0.7512E-01 - 0 . 2 7 9 7 E + 00 -0.5206E+00 - 0 . 2 9 9 8 E + 00

0 . 4 6 5 4 E + 00 0 . 4 5 8 1 E + 00 -0.5225E-01 - 0 . 6 0 4 8 E + 00 - 0 . 6 4 8 0 E + 00

Re ~

p = 1

0.3072E+00 0.5393E+00 0.3885E+00 - 0 . 3 8 9 3 E + 00 -0.1393E+01

0.1517E+00 0.3053E+00 0 . 2 7 0 8 E + 00 -0.1560E+00 -0.7895E+00

- 0 . 3 0 4 7 E + 00 - 0 . 2 5 6 4 E + 00 0.1046E+00 -0.3705E+00 0.2017E+00

Im ~

0.1862E+01 0.1216E+01 -0.4896E+00 0 . 6 3 8 7 E + 00 0.5185E+01

0.7978E+00 0.5014E+00 - 0 . 3 4 6 7 E + 00 -0.1073E+00 0.1661E+01

0 . 5 3 4 0 E + 00 0 . 5 0 5 5 E + 00 - 0.5970E-01 -0.5589E+00 - 0 . 4 3 0 2 E + 00

R e ~,

p = 2

0.3072E +00 0.5392E+00 0.3885E+00 - 0 . 3 8 9 3 E + 00 -0.1393E+01

0.1517E+00 0 . 3 0 5 3 E + 00 0 . 2 7 0 8 E + 00 -0.1560E+00 -0.7895E+00

- 0 . 3 0 4 7 E + 00 - 0 . 2 5 6 4 E + 00 0.1046E+00 0.3705E+00 0.2017E+00

Im

W a v e f u n c t i o n s e x p a n d e d f o r r < R a n d different k - v a l u e s r e t a i n i n g o n l y the first p o l e in the e x p a n s i o n ( r e c t a n g u l a r well: V o = 4 8 . 7 2 M e V , R = 3 f m , i.e. ~o = x o R = 4.6)

TABLE 6 a

2

;> Z

r

0.60 1.20 1.80 2.40 3.00

0.60 1.20 1.80 2.40 3.00

0.60 1.20 1.80 2.40 3.00

k

1.0

3.0

5.0

-0.7065E-03 0.14 i 3E - 02 -0.2119E-02 0 . 2 8 2 6 E - 02 - 0.3532E-02

-0.2259E +00 0.1968E+00 0 . 5 4 4 6 E - 01 - 0.2443E + 0 0 0. 15 83 E+ 00

0.5301E +,00 0.4824E + 00 -0.9101E-01 -0.5652E+00 -0.4234E +00

Re ~b

Exact

0.1248E-04 - 0.2496E - 04 0.3744E-04 - 0 . 4 9 9 2 E - 04 0.6240E-04

-0.1636E+00 0 .1425E+ 00 0 . 3 9 4 3 E - 01 -0.1768E+00 0.1146E+ 00

- 0.2930E+00 - 0.2667E + 00 0.5031E-01 0 .3125E+ 00 0.2341E+00

Im ~k

-0.2832E-01 0.7930E - 02 0.7374E-01 - 0 . 7 7 6 2 E - 01 -0.1144E+01

- 0 . 2 3 2 8 E + 00 0.1998E+00 0 . 7 2 6 2 E - 01 - 0 . 2 7 3 9 E + 00 - 0.2019E+00

0.5294E+ 00 0.4827E + 00 -0.8931E-01 - 0 . 5 6 8 4 E + 00 - 0.4614E+00

Re ~b

p = 1

- 0 . 8 5 9 7 E - 02 0.2268E - 01 -0.1615E-01 - 0 . 1 1 5 3 E + 00 -0.1306E+00

-0.1645E+00 0.1458E+00 0 . 3 7 8 6 E - 01 -0.1945E+00 0.9629E-01

-0.2931E+00 - 0.2666E + 00 0.5025E-01 0.3119E+00 0.2335E+00

Im ~b

-0.1214E-01 - 0.2115E - 03 0.3199E-01 0 . 1 1 4 4 E - 02 -0.2015E+00

-0.2270E +00 0.19 6 9 E+0 0 0 . 5 7 5 9 E - 01 -0.2455E +00 0.13 7 6 E+0 0

0.530 0 E+0 0 0.4824E + 00 - 0 . 9 0 9 8 E - 01 -0.5653E+00 - 0.423 7 E+0 0

Re ~,

p = 2

-0.8597E-02 0.2268E - 01 -0.1615E-02 -0.1153E+00 -0.1306E+00

-0.1645E+00 0.1458E+00 0 . 3 7 8 6 E - 01 -0.1945E+00 0.9628E-01

-0.2931E+00 - 0.2666E + 00 0.5025E-01 0.3119E+00 0.2335E+00

Im ~b

Wave functions expanded for r < R and different k-values with the first seven poles in the expansion (rectangular well: Vo = 48.72 MeV, R = 3 fm, i.e. flo = x o R = 4.6)

TABLE 6b

Z

X > 7~

0.5301E + 00 0.4824E + 00 -0.9101E-01 -0.5652E+00 - 0.4234E + 0 0

-0.2259E+00 0.1968E + 00 0 . 5 4 4 6 E - 01 - 0.2443E + 00 0.1583E+00

-0.7065E-03 0.1413E - 02 -0.2119E-02 0 . 2 8 2 6 E - 02 -0.3532E-02

0.60 1.20 1.80 2.40 3.00

0.60 1.20 1.80 2.40 3.00

0.60 1.20 1.80 2.40 3.00

1.0

3.0

5.0

Re tp

r

k

Exact

0.1248E-04 - 0.2496E - 04 0.3744E-04 - 0 . 4 9 9 2 E - 04 0.6240E-04

-0.1636E+00 0.1425E + 00 0 . 3 9 4 3 E - 01 -0.1768E+00 0.1146E+00

- 0.2930E + 00 - 0.2667E + 00 0.5031E-01 0.3125E+00 0.2341E+00

Im ~O

-0.1299E-02 0.8585E - 03 0.1258E-03 0.1234E - 01 - 0 . 3 0 2 4 E + 00

-0.2261E +00 0.1966E + 00 0,5524E - O1 - 0 . 2 4 0 9 E + 00 0.5184E-01

0.5300E + O0 0.4824E + 00 -0.9093E-01 - 0.5649E + 00 -0.4352E+00

Re @

p = 1

0.1127E-04 - 0.2381E - 03 -0.3653E-03 0 . 1 4 5 0 E - 02 -0.3256E-02

-0.1636E+00 0.1424E + 00 0.3934E - 01 -0.1765E+00 0.1139E+00

- 0.2930E + 00 - 0.2667E + 00 0.5031E-01 0.3125E+00 0.2341E+00

lm @

-0.7344E-03 0.1393E - 02 - 0 . 2 0 0 1 E - 02 0.3186E - 02 -0.8251E-02

- 0 . 2 2 5 9 E + 00 0.1968 E + 00 0 . 5 4 4 8 E - 01 - 0.2442E + 00 0.1577E+00

0.5301E + 00 0.4824E + 00 -0.9101E-01 -0.5652E +00 -0.4234E +00

Re ~

p = 2

4.6))

0.1127E-04 - 0.2381E - 03 -0.3653E-03 0 . 1 4 5 0 E - 02 -0.3256E-02

0.1636E+00 0.1424E + 00 0.3934E - 01 -0.1765E+00 0.1139E+00

- 0.2930E + 00 - 0.2667E + 00 0.5031E-01 0.3125E+00 0.2341E+00

lm qs

W a v e functions e x p a n d e d for r < R and different k - v a l u e s with the first 22 poles in the e x p a n s i o n ( r e c t a n g u l a r well: Vo = 48.72 MeV, R = 3 fm (flo =

TABLE 6C

> Z

N~

TABLE 7 b

22

N

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

k

0.6079E + 0 0 0.7473E + 0 0 - 0.8033E + 00 -0.9870E+00 - 0.9220E + 0 0 - 0.5073E + 00 -0.4155E+00 - 0.9603E - 01 0.3246E - 01 0.1891E+00

0.6079E + 00 0 . 7 4 7 3 E + 00 - 0.8033E + 00 - 0.9870E + 00 - 0.9220E + 00 - 0.5073E + 00 -0.4155E+00 - 0.9603E - 01 0.3246E - 01 0.1891E+00

Re S

Exact

0.7940E + 00 -0.6645E +00 -0.5956E +00 0.1608E + 00 0.3872E + 00 0.8618E + 00 0.9096E + 00 0.9954E + 00 0.9995E+00 0.9820E+00

0 . 7 9 4 0 E + 00 - 0 . 6 6 4 5 E + 00 -0.5956E +00 0.1608E+00 0.3872E+00 0 . 8 6 1 8 E + 00 0 . 9 0 9 6 E + 00 0 . 9 9 5 4 E + 00 0.9995E+00 0 . 9 8 2 0 E + 00

Im S

0 . 6 0 7 5 E + 00 0.7539E+00 - 0 . 8 3 6 3 E + 00 - 0 . 8 8 5 0 E + 00 - 0 . 1 1 6 4 E + 01 - 0 . 2 4 6 8 E - 01 - 0 . 1 2 7 0 E + 01 0 . 1 2 8 7 E + 01 -0.2051E+01 0.3147E+01

0 . 6 0 6 5 E + 00 0 . 7 6 9 6 E + 00 -0.9151E+00 -0.6383E +00 -0.1758E+01 0.1189E+01 -0.3476E+01 0 . 4 9 6 9 E + 01 -0.7803E+01 0.1166E+02

Re S

p = 2

0 . 7 9 6 9 E + 00 -0.6871E +00 - 0.5233E+00 0 . 1 9 9 7 E - 02 0 . 6 6 6 3 E + 00 0 . 4 4 3 0 E + 00 0 . 1 4 6 0 E + 01 0 . 3 6 2 2 E + 00 0.1613E+01 0.5538E+00

0 . 8 0 3 3 E + 00 - 0 . 7 3 7 1 E + 00 - 0 . 3 6 2 8 E + 00 -0.3511E+00 0.1287E+01 - 0.4828E +00 0.2651E+01 - 0 . 9 2 2 9 E + 00 0.2619E+01 0.5127E+00

Im S

0.6079E + 00 0.7473E+00 -0.8036E+00 - 0 . 9 8 6 0 E + 00 - 0 . 9 2 4 4 E + 00 - 0 . 5 0 1 8 E + 00 - 0 . 4 2 6 3 E + 00 - 0 . 7 6 5 0 E - 01 - 0 . 3 4 9 0 E - 03 0.2408E+01

0.6078E + 00 0.7485 E + 00 -0.8102E+00 - 0 . 9 6 2 2 E + 00 -0.9915E+00 -0.3410E+00 - 0.7703E +00 0 . 5 9 7 2 E + 00 -0.1229E+01 0 . 2 3 4 6 E + 01

Re S

p = 3

= 4.6)

Im S

xoR

0 . 7 9 4 0 E + 00 - 0 . 6 6 4 5 E + 00 - 0 . 5 9 5 6 E + 00 0 . 1 6 0 9 E + 00 0 . 3 8 7 0 E + 00 0 . 8 6 2 5 E + 00 0 . 9 0 7 4 E + 00 0 . 1 0 0 1 E + 01 0.9869E+00 0.1007E+01

0 . 7 9 4 0 E + 00 - 0 . 6 6 4 6 E + 00 -0.5948E+00 0.1582E+00 0.3916E+00 0.8621E+00 0.8792E+00 0 . 1 1 2 5 E + 01 ~).6104E + 00 0.1967E+01

T h e S - m a t r i x e x p a n s i o n f o r d i f f e r e n t k w i t h the first 7 a n d 22 p o l e s in t h e e x p a n s i o n ( r e c t a n g u l a r well : Vo = 4 8 . 7 2 M e V , R = 3 fm, i.e. flo =

Z

0.4334E 0.2427E -0.2975E - 0.4093E 0.6830E

- 0.2083E + 00 - 0.2445E-01 0.2055E + 00 0 . 4 8 5 6 E - 01 -0.1998E+00

0.60 1.20 1.80 2.40 3.00

0.60 1.20 1.80 2.40 3.00

1.5

2.0

+ 00 + 00 +00 + 00 - 01

0.4958E + 00 0.4513E+00 -0.8513E-01 -0.5287E+00 -0.3961E+00

0.60 1.20 1.80 2.40 3.00

1.0

+ 00 + 00 + 00 +00 + 00

0.7799E 0.8847E 0.2238E - 0.6309E - 0.9394E

0.60 1.20 1.80 2.40 3.00

0.5

Re ~b

•

k

Exact

+ 00 + 00 +00 + 00 - 01

0.4341E+00 0.5095E-01 - 0.4282E + 00 - 0.1012E + 00 0.4163E+00

0.2612E 0.1463 E -0.1793E - 0.2467E 0.4117E

- 0.8791E - 01 - 0 . 8 0 0 0 E - 01 0.1509E-01 0.9374E-01 0.7022E-01

- 0.3973E + 00 - 0.4507E + 00 - 0 . 1 1 4 0 E + 00 0.3213E+00 0.4786E + 00

Im 0 + + + + +

00 00 00 00 00

-0.1455E+00 -0.9787E-01 0.1875E + 00 0.2243E + 00 -0.4262E +00

0.4640E + 00 0.2113E + 00 -0.3138E+00 - 0.3240E + 00 - 0.2488E - 01

0.5079E + 00 0.4396E+00 -0.9301E-01 - 0.4949E + 00 - 0 . 4 3 0 0 E + 00

0.7927E 0.8821E 0.2218E - 0.6229E - 0.9470E

Re ~,

p = 1

+ 00 + 00 +00 + 00 - 01 0.4844E+00 0.3203E-01 - 0.4955E + 00 - 0.2061E - 01 0.4889E+00

0.2755E 0.1432E -0.2021E - 0.2252E 0.7492E

- 0.8459E - 01 -0.8041E-01 0.9279E-02 0.9848E-01 0.8018E-01

- 0.3969E + 00 - 0.4507E + 00 - 0 . 1 1 4 6 E + 00 -0.3219E+00 0.4798E + 00

lm ~

-0.1894E+00 -0.5736E-01 0.2195E + 00 0.1002E + 00 -0.3093E+00

0.4393E + 00 0.2341E + 00 -0.2958E+00 - 0.3938E + 00 0.4084E - 01

0.4969E + 00 0.4498E+00 -0.8500E-01 -0.5259E+00 - 0.4008E + 00

0.7799E + 00 0.8346E + 00 0.2238E + 00 - 0.6307E+00 - 0.9397E + 00

Re ~b

p = 2

+ 00 + 00 +00 + 00 - 01 0.4844E + 0 0 0.3203E-01 - 0.4955E + 00 - 0.2061 E - 01 0.4889E+00

0.2755E 0.1432E -0.2021E - 0.2252E 0.7492E

- 0.8459E - 01 -0.8041 E-01 0.9279E-02 0.9848E-01 0.8018E-01

- 0.3969E + 00 - 0.4507E + 00 - 0 . 1 1 4 6 E + 00 0.3219E+00 0.4798E + 00

Im

W a v e functions e x p a n d e d for • < R a n d different k - v a l u e s w i t h the f o u r poles in the e x p a n s i o n ( r e c t a n g u l a r well: Vo = 48.72 M e V , R = 3.26 fm, i.e. flo = 5.0)

TABLE 8

Z

Z

X

© r-"

416

J. B A N G et al.

N:,/ i0 0 '

S exact S p

.N=22 / N =/.3 , N = 6/..

lO ~

16' = 22

///// = /,3

-- 6 z ,

162

16~

0

10-3

K I

2

3

4

5 f m -~

Fig. 2a. The value of IS~a°'-Sp)/Sexa¢' I for a r e c t a n g u l a r well with/~o = 4.6 and p = 2.

l

2

3

4

s fm--~

Fig. 2b. The s a m e as fig. 2a for p = 3.

the values I(S~X"ct-SP)/S. . . . t I f o r p = 2 (fig. 2a) a n d p = 3 (fig. 2b),/~o = 4.4. Fig. 3 shows the same values for/~0 = 5.0, p = 2 (dotted curve) and p = ~ (full drawn). Table 6 shows the exact wave function and its expansion with p = 1, 2 and n % l, 7, 22 for/~o = 4.6. In table 8 corresponding results with n < 4, for/~o = 5.0 are shown. F r o m the tables and figures it is seen that the convergence is strongly dependent on p and becomes faster for larger p-values. This is the advantage of the Mittag-Leffler expansion, as p can be chosen so as to get a fast convergence. F o r these tables, it is seen, that for n < 22, the Mittag-Leffler expansion expression for the wave function is correct with three to four digits for k between 0 and 3 fm 1 and in some cases for larger k. A similar g o o d convergence ls seen for the Green functions, as in the f-potential case.

POLE E X P A N S I O N

,°,I

417

se"actsP)'I Sexact

10 0

t61

2, p=3

162

163 v

Fig. 3. The value of

~

rill

]S~Xact-SP)/S ex'ct] a s a

function of k for different Mittag-Leffler expansions for a rectangular well with fl0 = 5.0.

F r o m the general relations given above [-see (7) and below] it only follows that the Mittag-Leffier expansion of the wave runction converges for 0 < r < R, however, it is seen that convergence is also good for r = R. If this point is regarded as the limit r ~ R - 0 , this is not surprising. 5. Conclusions We have shown here that the Mittag-Leffier expansion method can be used to calculate simple, analytic expressions in k for the S-matrix, scattering states and Green functions of the problem of one particle in a potential of a sort which includes, with reasonable accuracy the usual nuclear potentials. It should be stressed that although the expansion + = ~ a,q~,(r) .

k-k.

(5.1)

418

J. B A N G et al.

is valid only for r < R, an equally exact expression for ~9+ is obtained for r > R, using the Mittag-Leffler expansior~ of the S-matrix. These expansions converge uniformly in all the k-space, except at the poles, and in practice, the convergence can be improved, by choosing the constant p sufficiently large. In the case where only one pole needs to be taken into account in the expansion of ~9+, we obtain the factorised expression tO+ ~ ~o,(r)F(k),which was used earlier by several authors in nuclear reaction calculations 25). As shown by some examples in sect. 3, the Mittag-Leffler expansions highly facilitate the inclusion of continuum states in nuclear structure calculations, e.g. of the shell model type. The same will be the case with other nuclear structure calculations, e.g., with RPA methods. However, the non-orthogonality of the wave functions means that some care is needed, when they are used in such problems. Other discrete basis sets can be used in such calculations, e.g., the above-mentioned functions of K apur and Peierls, etc., or the harmonic oscillator functions. Nevertheless, the expansion in resonance states has the advantage of giving a much simpler expression for G+(k) in the complex k-plane, thus facilitating the calculation of matrices like (3.18). Also, the particle emission properties of mixed states are in such a calculation determined in a simple way through S(k). Such structure calculation will be the subject of a later publication. In conclusion, the authors want to thank collaborators of JINR, Dubna and the Niels Bohr Institute, Copenhagen, particularly B. Nilsson, for enlightening discussions. One of the authors (J.B.) wants to thank JINR for paying for his stay in Dubna for a period in which part of the work was done, and the Danish Scientific Research Council for a grant covering the travel expenses of this stay. Appendix

It is easy to prove that the real poles are single, but somewhat more complicated to exclude multiple complex poles. This is often done in the literature by arguing that such poles are a very rare phenomenon, which will only be seen for particulm potentials. This argumentation seems in need of a firmer, quantitative basis, which we shall attempt here to construct. For all potentials with parameters, which can be varied in such a way that the poles are moving, the complex poles will become real by meeting pairwise at 0, for l 4 0, or on the negative imaginary k-axis, for l = 0. These double poles, which can be avoided by a slight change of the parameters of the potentials, are, for the potentials of relevance in nuclear physics, essentially the only ones. The resonance condition (2.8) will lead to a multiple resonance if d f!d/dr)tp~(r)] d dk L q~(r) 1,=~1~=~, - dk

fkh:'(km-]l L h -(kR)

(A.1)

POLE EXPANSION

419

In this case, the normalization condition (2.9) cannot be fulfilled, since (A.1) leads to the identity d,+

\

(A.la)

= o.

Let us now consider a square well 0_,

g(r) =

for r > R for r = R

go,

R = 1,

(A.2)

then tp = const x jl(Kr),

(A.3a)

K = (k 2 + Vo)½.

(A.3b)

with

The resonance condition (2.8) is now d + ,. d kd-k [In (h, (k))]~k=kx, = K ~ [In (jI(K))]K=K(ka),

(A .4)

and the condition (A.1) that it is a multiple pole is d d 3 d [ k d ( l n ( h ~ ' ( k ) ) ) l k = k ~ - d-k [ K ~ (In (Jt(K))).J'k = u~(t(= x (k);

(A.5)

or

d2 d k ~-~ (In (h~+ (k))) + (ln

(h~ (k)))

d2 K dK k d = K~ (ln (j,(K))) + k dk K dK (ln (jr(K))),

K -- K(k),

k

ka. (A.6)

Now, we can use the Riccatti-Bessel equation, satisfied by z = h + and z = Jz: d2 dp2

(In (z(p))) +

(~pp

(In (z(p)))

)2

l(l+l) p2

1,

(A.7)

K dK k- d---k= 1,

(g.8)

[seen from (A.3b)], to transform this into d

[

k2

I-

(ln(hi~(k)))

-

2

K2

d k d + ~ (In (h~-(k)))- ~ -~ (ln (j~(K))) = 0,

K = K(kO,

k = k;,,

(A.9)

J. BANG et al.

420

or, using (A.4), Vo [ d + 1(1+ 1) ~K ] dk (In (h~ (k)))+ k

d

+

(A.IO)

or

d

d

dr (ln (~ox))k=~ = k ~ (ln (h[(k)))lk~k~=

{ -- l

I+1'

(A.11)

which could not be fulfilled for a complex pole, since e.g.,

2 Im I d ( l n (¢P~))]r= ,

= -rz

f/

r,,

12dr

[tp(1)12 < 0,

(A.12)

for a real potential. Now with the normalization (2.9), the shift of a pole position, 6Ez for a small change of the potential, 6V, is given by the usual perturbation expression 6E a =

f

(A.13)

[cf. eq. (3.29)]. This perturbation approach, which presupposes that the poles are single, will converge and lead to a new potential with non-degenerate eigenvalues, provided [•Ei[ is smaller than [Ei - Ei[ for all Ej. In nuclear physics, the Woods-Saxon potential is the most used. This is strictly speaking different from zero for all r-values. For the present purpose it is, however, completely legitimate to replace it by zero outside R, provided this is chosen well beyond the nuclear radius. This potential will also, for all practical applications, deviate from a square well by a 6V which fulfills the above requirement.

References 1) C. Mahaux and H. A. Weidenmiiller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, 1969) 2) L. Eisenbud and E. P. Wigner, Phys. Rev. 72 (1947) 29 3) P. L. Kapur and R. E. Peierls, Proc. Roy. Soc. A166 (1938) 4) K. Meetz, J. Math. Phys. 3 (1962) 690 5) T. Sasakawa, Nucl. Phys. A160 (1971) 321 6) J. M. Bang and F. A. Gareev, Preprint JINR, E2-10624, 1977 7) J. M. Bang and F. A. Gareev, Preprint JINR, E2-10625, 1977 8) G. Gamow, Z. Phys. 51 (1928) 204 9) R. E. Peierls, Proc. Roy. Soc. A253 (1959) 16 10) J. Humblet and L. Rosenfeld, Nucl. Phys. 26 (1961) 329

POLE EXPANSION 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)

Ya. B. Zel'dovich, JETP (Sov. Phys.) 12 (1961) 542 T. Berggren, Nucl. Phys. AI09 (1968) 265 W. Romo, Nucl. Phys. A l i a (1968) 618 G. Garcia-Calderon, Nucl. Phys. A261 (1976) 130 G. Garcia-Calderon and R. Peierls, Nucl. Phys. A265 (1976) 443 M. M. Nussenzweig, Causality and dispersion /"elations (Academic Press, New York, 1972) B. I. Serdobol'skij, JETP (Soy. Phys.) 36 (1959) 1903 J. R. Taylor, Scattering theory (John Wiley, New York, 1972) V. de Alfaro and T. Regge, Potential scattering (North-Holland, Amsterdam 1965) J. Block et al., Rev. Mod. Phys. 23 (1951) 147 U. Fano, Phys. Rev. 124 (1961) 1866 J. Bang, F. A. Gareev, J. V. Puzynin and R. M. Jamalejev, Nucl. Phys. A261 1976) 59 R. M. More and E. Gerjuoy, Phys. Rev. A7 (1975) 1188 H. A. Weidenmiiller, Ann. of Phys. 28 (1964) 60 J. Bang and J. Zim~inyi, Nucl. Phys. A139 (1969) 534 W. Romo, Nucl. Phys. A237 (1975) 275

421