On the theory of complex potential scattering

ANNALS

OF PHYSICS

131,42W50 (1981)

On the Theory

of Complex

Potential

Scattering

L; P. KOK Institute for Theoretical Physics, P.O.B. 800, Groningen, The Netherlands AND

H. VAN HAERINGEN Department of Mathematics,

Delft University of Technology, De&

The Netherlands

ReceivedJuly 17, 1980

Westudytheeffectof theadditionof a complexpotentialhVsep to anarbitrary Schrodinger operatorH = I& + V on the singularities of the S matrix, asa function of X. Here V,,, isa separable interaction,andh is a complexcouplingparameter.The pathsof thesesingularitiesare determinedto a great extent by certain saddle points in the momentum(or energy)plane.We explain certaincritical phenomenarecentlyreportedin the literature. Associatedwith thesesaddles are branch-typesingularities in the complexAplane,which are dynamicalin origin. Someexamplesare discussed in detail.

1.

INTRODUCTION

In quantum mechanics many physical phenomena and processes are described using the concept of a potential. Usually the potential functions are taken real, and the corresponding Hamiltonians hermitian. In order to describe annihilation and absorption in certain processes it is often convenient to work with models in which the potential is no longer real. We mention exotic atoms, where in addition to the long-range Coulomb potential short-range annihilation forces are acting and the nucleon-antinucleon system, where the strong nuclear force of short range contains an annihilation component, of even shorter range. Perhaps the most widely known complex potential is that of the optical model in particle and nuclear physics. In fact, in any process with more than one open channel, the effect of one or more of the additional channels can be taken into account formally by the introduction of a complex potential. For real potentials the quantum mechanical theory of potential scattering is well developed, and various textbooks on this subject exist [I]. For complex potentials the theory can be expected to be more complex. In this paper we wish to study the effect of the addition of a complex potential to a given Hamiltonian on the bound-state and resonance positions. This is a complicated problem. We therefore investigate first the somewhat simpler problem of

426 0003-4916/81/02042~25$05.00/0 Copyright All rights

0 1981 by Academic Press, Inc. of reproduction in any form reserved.

COMPLEX

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427

the addition of a (hermitian) separable potential, multiplied by a complex coupling parameter X. The movement of bound-state, resonance, virtual-state (and other) poles as a function of real coupling parameters has been discussed extensively in the literature (see, for example, the work in Ref. [l], and papers by McVoy et al. [2]), The effect of the addition of a rank iV separable potential to a given SchrGdinger operator, as a function of a real coupling parameter h has been investigated by us in a previous paper [3]. This work will now be extended to the case of a complex coupling parameter. It turns out to be convenient to recall some well-known facts about analytic functions of a complex variable, which can be found in the standard textbooks. Particularly useful in this context is a monograph by Lauwerier [4], from which we quote freely in the first part of Section 3. This investigation is motivated by certain at first sight unexpected phenomena, occurring in model calculations (involving complex potentials) for various physical systems. As a function of the complex potential strength parameter h Simonov and co-workers [5] observed the existence of a so-called critical value of arg h. When arg h passes through this critical value the trajectories of bound-state (and of quasibound-state, resonance, etc.) poles as a function of 1h / undergo a sudden and dramatic change. The reason for this is explained in this paper. We also discuss the phenomenon of reconstruction of (Coulombic) spectra [6], and of the oscillatory behavior of width and energy of levels of exotic atoms under variation of the strength of the nuclear interaction [7]. Both these phenomena have been understood and explained before. Also, we reexamine the effect of an absorptive potential on bound states and resonances as studied by Myhrer and co-workers [S]. Reference [9] contains related work carried out by Bailey and Schieve, who give further references concerning complex energy eigenstates in quantum decay models. The layout of this paper is as follows. In Section 2 we present the necessary formulas which give the map between h and the pole positions. In Section 3 general features of the pole trajectories are discussed. Section 4 considers some specific cases. More general complex potentials are treated in Section 5. Finally, Section 6 further extends and discusses the results, whereas Section 7 concludes the paper.

2. PRELIMINARIES

We want to study the effect of the addition of a separable interaction

vs= --h Ig>(gI

(2.1)

to a given Schrodinger operator H=H,+V

(2.2)

in three-dimensional space, with H,, = k2 the kinetic energy operator. We have taken fi = 1, 2~ = I. The interaction V can be a local potential V(r) vanishing for

428

KOK

AND

VAN

HAERINGEN

r + co, or a separable potential. We require V to be well behaved, without giving, however, a precise mathematical definition. The form factor 1g) in Eq. (2.1) is assumed to have a finite norm, (g 1g) < co. The operator Va is acting in a space corresponding to one particular value of the angular momentum quantum number 1. In fact we shall restrict ourselves to 1 = 0, although the results are easily generalized to 1 > 0. We are interested in the behavior of the bound-state eigenvalues of HA 3 H + V, as a function of the coupling parameter A. We shall consider not only real values of A. It turns out to be useful and illuminating to consider A as a complex parameter. When h is complex it may be confusing to talk about bound states, and therefore we shall usually speak about poles of the amplitude, or of the S matrix. These poles on the so-called physical sheet of the E plane, or on the physical half of the k plane, can indeed for real X be associated with bound states. We shall denote the eigenstates of H at the nonpositive eigenvahres -Key (n = 1, 2,...), which we assume to be nondegenerate, by 1K,). We limit ourselves to those cases in which there are no (isolated) positive eigenvalues. The (so-called outgoing) scattering states of the unperturbed Hamiltonian H corresponding to the continuous spectrum are denoted by I kf). The orthonormality properties of the eigenstates and the scattering states are

(k + 1k’f)
= k-a S(k - k’),

(2.3)

= 0.

Completeness then is given by the following resolution of the identity 1 = c I K,)(‘% n

I + j-= dk k2 I k+Xk+

I.

0

(2.4)

The Green operator or resolvent G(E) = (E - H)-l

(2.5)

can be decomposed as (2.6)

The eigenvalues of HA are just given by the poles of the resolvent G,(E) = (E - H&l.

(2.7)

Obviously on the physical sheet G,(E) can have poles only for red nonpositive values of E, as long as X is real. (Or, considered as a function of k, in the physical half of the k plane G,(k2) can have poles only on the positive imaginary k axis, or at k = 0, as long as h is real.) ’

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SCATTERING

It is not difficult to derive from the following expression for GA ,

GlgXgl

G

(2.8)

GA=G-X-l+(g~G~g) the interesting equality

(g I G1\I g>-’ = X + (g I G I d-l. The poles of G and GA at E < 0 (at k = in, at such a pole is separable, for example

K

> 0) are all simple poles. The residue

$pA (E - EnA) G,(E) = 1 KnA)(KnA

n

Here /

(2.9)

1 =

P,

.

(2.10)

is the eigenstate of HA corresponding to the nondegenerate eigenvalue From Eq. (2.8) it is clear that to determine the poles of G,, it is sufficient to investigate the poles of G, and the zeros of h-l + (g / G I g). In what follows we shall assume that / g) is not orthogonal to any bound-state vector / K,J of H. (In fact, if j g) and some I K,) were orthogonal we would have a fixed pole, which is stable against variations of h. We consider such a fixed pole a coincidence, which in turn can lead to accidental degeneracies. Its inclusion in the formalism would be quite simple as was shown for real h in Ref. [3].) From Eq. (2.6) it is clear that the expression (g I G(- K”) I g) (we take E = -K2 < 0) has simple poles at K = K, . Also G and G ] g) have simple poles at these values of K. From Eq. (2.8) we have EnA =

K,‘)

-(K%~)~.

&I

(-

K2

+

Kn2)

GA(-

K”)

n

(2.11) This implies that GA has no poles at from the solutions of the equation

K =

K,

The poles of G,+are therefore obtained

.

h-l + (g j G(-K~)

I gj

= 0.

(2.12)

Equation (2.9) confirms that GAhas no pole when G has a pole (unless h = 0, of course). Relation (2.12) is the basic relation of this paper. It can be rewritten in the form h-l

=

(2.13)

n(K)-‘,

or in the form h

where the function

(1(K)

(2.14)

=A(,),

is defined through its reciprocal (1(K)-1

=

(g

1 (K2

+

H)-’ / g).

(2.15)

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KOK

AND

VAN

HAERINGEN

As a function of the complex variable K, (1 is real analytic, n(K*)

According to Eq. (2.6) the function /q,)-l

=

(2.16)

n*(K).

can be represented as

A(K)-’

= c K= K,= n

Its derivative with respect to f

K

(2.17)

is

(1(K)-1

=

-2K(

g 1 (K2

+

ff)-’

(2.18)

1 g).

For real K > 0 this is always negative, so that /l(K)-’ is a monotonic decreasing function on the positive K axis on each interval between (or outside) the poles K~ , K2 ,.. ., Kn, that is, O
KN

,...,

K2

<

K

<

K1,

K1

<

K

<

CO.

(2.19)

Since /1(~)-l takes all values between + 03 and -co between two consecutive poles once, there is one solution of (2.13) (and consequently one solution of (2.14)) at each such interval for real X. If X is positive (I’s is attractive) there is an additional solution for K~ < K < CO,and possibly one for 0 < K < KN . If h is negative (Vs is repulsive) there may be one more SdUtiOn for 0 < K < KN only. Summarizing, one may say that as a function of h the bound state poles of H,, can move only between the zeros Of /1(K)-l on the positive K axis if h varies between - co and + co. These zeros alternate with the poles. (In this context it is sometimes convenient to consider also K = co as a zero). All poles are therefore limited in their freedom of movement. Only the ground-state pole has a somewhat larger freedom as it can move towards fee for large h. The reason for the limited freedom of movement is the rank-one character of Vs . If V = 0, and therefore His the kinetic-energy operator, it follows that there can be at most one bound state, for X sufficiently positive. This is well known. We conclude this section with the derivation of the bound-state vectors of H,, , using Eqs. (2.Q (2.9), and (2.10). Taking the residue of G,(E) at E = EnA one finds that the bound-state vector corresponding to the negative eigenvalue E,,A = -(K,,~)~ of HA is given by

I 6~‘) = ~G(Ena~I g>,

A- = (g I G2@nA)I g>.

From Eqs. (2.9) and (2.20) the orthonormality
are easily derived.

/ K;‘)

(2.20)

properties of these bound-state vectors =

6,,,

(2.21)

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SCATTERING

3. POLE TRAJECTORIES In Section 2 it was shown that the poles of G, are found from the solution of the relation (2.12), or alternatively from the solutions of Eq. (2.14), which we repeat here for convenience, x = A(K).

(3.1)

It is useful to recall some simple properties of analytic functions. Let A(K) be an analytic function of the complex variable K = x + iy. In terms of its real and imaginary parts we have /1(K)

The property of analyticity

=

Re /I(x, y) + i Tm L&x, y).

0.2)

may be expressed by the Cauchy-Riemann

aRe/l -=-

aImA aY

2X

aReA -=--. ay

’

aImA f3X

relations (3.3)

In fact, the analytic function A(K) may be described in a rather complete way by the landscape / A( or by (Re (1)” + (Im /I)“. In order to make the description unique one needs to know the phase of A(K) at a single point only. Stationary points of this modular landscape (where the tangent plane is horizontal) are given by A’(K) = 0. At a stationary point Kg the local situation is determined by the first few terms of the Taylor expansion, (1(K)

=

(1(K,,)

+

;(K

-

K,J)~A”(K~)

+

.”

.

(3.4)

It is easily concluded that the modular landscape of/l(K) has saddles at all stationary points Kg where (l(KO) # 0. A zero Ofd(K) of course alWayS gives an absolute minimum of 1n(K)/. Apart from this there may be singular points when j A( = co. We are interested in the complex transformation given in Eq. (3.1). By this transformation the steepest descent lines of the modular landscape, and the level curves (which locally are perpendicular to each other) are transformed into straight lines through the origin (rays) and circular lines with the origin as center in the complex h plane. The transformation (3.1) is singular at the saddles ofA( Any point where A’(K) = 0 gives rise to a branch point and a corresponding branch cut in the complex h plane. An arbitrary path C, in the complex K plane is transformed by the map (3.1) into a path C, in the complex h plane which meanders between the branch points and poles. From these branch points we have branch cuts going to infinity. We need not worry about the fact that C, passes through branch cuts, or crosses itself, when we take properly into account the structure of the Riemannian h plane. We now may locally deform the contour C,, in the h plane. To this corresponds a deformation of the corresponding contour C, . As long as C, during its deformation does not cross branch points the contour C, will not cross saddle points. In particular

432

KOK AND VAN HAERINGEN

(a) FIG.

lb1

1. (a) Contour C,+in the complexXplane;(b) contour C, in the Kplane.

if C, is a contour with begin and end points corresponding

to h = 0, and 1 h ] = co, respectively, its begin and end points in the complex K plane will remain the same. If, on the other hand, C,, does cross a branch point (dotted line in Fig. la) when locally deformed, the contour C, will have a different end point (or a different starting point) and therefore in general a completely different course (Fig. lb, tail sweeping). It thus appears that a local change of the trajectory C, may lead to a global change in the trajectory C, . To complete the picture one has to keep in mind the following. To one trajectory C, in the K plane corresponds one trajectory C,, in the X plane (which is uniquely determined by C, , as is clear from Eq. (3.1)). However, to the one C,, there correspond many trajectories in the K plane. To be more specific, there will be as many trajectories as there are zeros of A(K). One of these is C, . Another is C: as indicated in Fig. 2. Making the small local detour around a branch point in the X plane (cf. Fig. 1) then leads to relatively small changes in the combined paths C, , C: . As indicated by the dotted lines, they exchange their end parts (Fig. 2, tail swapping). The saddle points in the K plane play an important role in the course of the trajectories C, . In general the saddle points can be determined by solving $

FIG.

A(K)

=

0,

Or

$

(1(K)-’

=

0

2. TrajectoriesC, and C: in the Kplane.

(3.5)

COMPLEX

POTENTIAL

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SCATTERING

for complex K. The solutions Kg then determine the branch points &, Eq. (3.1). Using Eq. (2.18) this condition leads to 2K(g

j (K”

+

H)-’

1 g>

=

(3.6)

0.

(In general this is not equivalent to (K~ + H)-’ / g) = 0. We note that K general does not correspond to a saddle point for zero angular momentum.) the resolution (2.17) of n(K)-’ condition (3.6) can be formulated as c

n


1 Kn>(Kn

1 g>

(K2 - ‘%“>”

+ j-m dk k2 o


cf.

=(1(~,,),

f

=

0 in Using

I g: = o

k”)2

If Kg is a saddle point so is K$ . (If A, is a branch point so is A,*.) It can be easily demonstrated that saddle points cannot occur on the positive imaginary k axis (i.e., on the positive real K axis). (The last few statements are only valid for hermitian H, cf. Eq. (2.2), and need to be revised when we try to include non-hermitian Hamiltonians HI. If in the complex K plane we encircle a saddle point the corresponding trajectory in the h plane encircles the associated branch point twice to return to the original Riemann sheet. In the rest of the K plane there are the other K trajectories which (in general) all but one encircle a point twice. Saddle points in the K plane are the only points where two poles can collide. So far we have discussed only saddle points of so-called second order, where a first derivative vanishes. When two poles collide at such a point they have a head-on collision, and then scatter by an angle n/2, see Fig. 3a. This follows easily from Eq. (3.4). Of course higher order saddles may occur. In an nth order saddle the first IZ - 1 derivatives vanish, and n poles collide at that point to be scattered by an angle of rr/n, see Fig. 3b. They lead to branch points in the complex X plane of order n. In complex potential scattering these situations (n > 2) virtually never occur spontaneously. Yet, in the practical description of neutron-deuteron scattering of Ref. [lo] nucleon-nucleon potentials have been selected, leading to nth order saddles, with values of PZup to 16. In the next section we make some general observations concerning the movement of poles in the k plane (or the K plane) and we discuss some examples.

ia)

lb)

FIG. 3. Saddle points in the K plane are the only points where poles can collide. (a) Two-pole collision, second order saddle; (b) three-pole collision, third order saddle.

434

KOK AND VAN HAERINGEN

4. MOUNTAINEERING

ON MODULAR

LANDSCAPES

Often it is advantageous to consider an analytic function of the kind exp (1(K) instead of (1(K) itself. The modular landscape is then given simply by exp Re (I(x, y), whereas the phase of the complex function then is ImA(x, JJ) (cf. Eq. (3.2), K = x + iy). The level curves of this new modular landscape are given by Re A(x, JJ) = constant, whereas lines of steepest descent have Im A(x, y) = constant. Let us rewrite the map (3.1) as ReX+iImX=ReA(x,y)+iIm~(x,y).

(4.1)

By this transformation the level curves and the steepest descent lines of exp /l(K) are transformed into straight vertical lines and straight horizontal lines in the complex h plane, respectively. (Recall that the level curves and steepest descent lines of the modular landscape of (1(K) itself are transformed into two other mutually orthogonal sets of curves in the complex h plane, namely 1X 1 = constant, and arg X = constant, circles and rays, respectively). The saddle points of the modular landscape OfA of course are also saddle points of the modular landscape of exp A(K). Care is needed when /1 = 0, and consequently 1A 1has a minimum. In the second example discussed in the remainder of this section such a situation occurs. There we see that at this point we have a saddle point of expA(~). This saddle point leads to a branch point in X = 0. Indeed the representation of h in terms of 1X 1and 4 5 arg h is not suitable at this point. Instead (Re A, Im A) is a convenient representation. We now will study the map (3.1) in some specific examples. This relationship between K and the coupling parameter A is studied for complex values of both variables. The domain of analyticity of /l(K) is clear in each of the examples. EXAMPLE 1. The simplest case we can consider is V = 0, so that H = Ho = k2 (cf. Eq. (2.2)), and a simple form factor. We shall take

(P’ I vs IP> = -kc

PM PI,

g( PI = cw7Y(

P2 + B2F2,

(4.2)

in spite of the fact that the norm of this form factor is infinite. In Section 2 we have excluded such a form factor. Indeed some difficulties are associated with the form factor in (4.2). These concern the eigenstate vectors. No anomalies occur in the eigenvalues considered as a function of h. The function A(K), given by Eq. (2.15), is simply

A(K) = (K + p)//%

(4.3)

and the solution of Eq. (3.1) is K =

hfl

-

@.

(4.4)

COMPLEX

POTENTIAL

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SCATTERING

I

-

IF

I

FIG. 4. Variouspoletrajectoriesin the Kplane(k plane)corresponding to absorptiveinteractions (4.2). The physicalhalf-planeis shaded.

For attraction (h real, $ 2 arg X = 0) with increasing h we obtain a zero energy bound state at h = 1. Upon increasing the coupling parameter the binding energy K’ increases, too. For repulsion (h real, arg x = 5~) with increasing ) x 1, K travels to - co, further into the unphysical region. For small absorption (arg X small positive) the trajectory of K for increasing ) x 1enters the physical half of the K plane no longer at K = 0, but at a point slightly above the origin. This is shown in Fig. 4, which depicts a number of trajectories C, , corresponding to I X j E [0, co), 0 < $ d r. Rotated by in this figure shows the k plane (k = iK). Just above the real positive K axis we have the region where poles represent “bound states with small decay widths.” EXAMPLE 2. We introduce one additional feature of interest by considering again V = 0, but now combined with the Yamaguchi form factor taken in Eq. (2.1), g(p)

=

2,-113p3qp3

+

(4.5)

jy.

In this case

A(K) = (K + b?“/p”,

(4.6)

and the two solutions of Eq. (3.1) are well known, K1

=

/3hW -

p,

tc2

=

-piw2

-

p.

(4.7)

At h = 0 both solutions start at K = -p (i.e., at k = -i/3). For attraction (h real, arg h = 0), with increasing / X / the two points move away from each other along the real K axis. At h = 1 the solution K1 enters the physical half-plane to represent the bound-state pole. It continues to travel to $-co for h ---f + co. For complex h = 1h I ei* the two solutions again travel in opposite directions along straight lines

436

KOK

AND

VAN

HAERINGEN

FIG. 5. Various pole trajectories in the K plane (k plane) corresponding (4.5). The physical half-plane is shaded.

to absorptive interactions

starting from the point K = -p. These directions have angles +/2 and (5/2 + rr with the real K axis, respectively. This is shown in Fig. 5, where as in Fig. 4 we show only trajectories for Im h > 0, i.e., for 0 < 4 < rr. For small positive 4 the separable interaction Va may be thought of having a small absorptive component. After entering the physical half-plane the solution K1 then may be thought of representing a bound state with small decay width. Unlike in the previous example the modular landscape of expd(K) has a saddle point, namely at K = -/3. (Note that, as remarked previously, the modular landscape of /l(K) has an absolute minimum corresponding to n = 0 at this point.) This indeed is the only point in the K plane where two trajectories C, corresponding to the same trajectory C, can meet, or rather, where poles can collide (with scattering angle 4~). This saddle point is far from the physical hall-plane. A much more interesting situation is created when saddle points occur in the physical region. This will happen in the following examples. Although it is both tempting and instructive to consider (in the line of the previous examples) the cases g(p) K (p” + p3-““3 n >, 3, we shall not do so here. EXAMPLE 3. In this example we shall consider V different from zero. We choose V to be a separable interaction of large range with real coupling parameter such that H = H,, + V has one bound state. For definiteness we assume the Yamaguchi form factor (4.5) with a range parameter /3’ which is much smaller than 8, and the dimensionless strength parameter h’ fixed and larger than 1 in order to support the bound state. Again n(K) has a simple form,

The solution of Eq. (3.1) gives four poles Ki , i = l,..., 4. For 1x 1 = 0 they are situated at K1 = /?h’l12 - p’ > 0 (the bound state), at K2 = ---/?x’lf2 - t!?’ < 0 (a typical

COMPLEX POTENTIAL

I

SCATTERING

437

F-

FIG. 6. Sketch of pole trajectories in the K plane for Example 3. The physical half-plane is shaded.

“left-hand cut” pole in the terminology of S matrix theory), and at K~ = K~ = -/?L (As in the previous example we have a saddle at K = -p.) For / h 1 = co, the four poles are situated as follows. Two poles, which we choose to name K~= and K~-, lie at infinity, the other two lie between the positions of K~ and K~ for I h / = 0. We call these ~~~ (slightly to the right of K~ (I h / = 0)) and ~~~ (slightly to the left of ~~(1 X / = 0)). The two small distances here are determined by the small parameter /3’/& This is illustrated in Fig. 6. We wish to study the trajectories of the poles K$ as a function of 1X j, where I h 1 is increased from 0 to co, for fixed values of 4 = arg h, 0 < I$ < V. Roughly speaking, all these cases correspond to absorptive short-range potentials. First consider 4 =_ 0 (Vs is attractive). Upon increasing h, Kq moves to the left and K~to the right, initially at the same (large) speed. K~ moves to the right and K2 to the left, slowly. Consequently K~ and K2 are heading towards each other. They cannot pass without noticing each other. Therefore they collide and scatter by an angle in. This happens at a saddle point of A(K) at the negative K axis. Scattered into the complex IC plane off the real aXiS the two poles remain conjugate (K2 = Kz) because of the real-analyticity of A(K), until they return to the axis, collide again and scatter again by -ir to continue their travel on the real axis (another saddle). Although in principle it is impossible to tell which pole goes where (up or down, left or right) in the scattering processes, one can state that this is their (rather cumbersome) way of passing each other when they meet on the real axis. Because for real h we cannot have conjugate pairs of poles in the physical half-plane, this process can only take place in the left half K plane (lower

438

KOK AND VAN HAERINGEN

half k plane). Pole K~ continues its travel to the right, enters the physical half-plane to represent a second bound state, and finally reaches Key. Pole KS then has reached ~3~, whereas K~ has moved to + 00, and K~ to - co. Note that K1 originally represented the shallow bound state of the “long-range” potential, but finally represents the deep bound state of the strong short-range (say, nuclear) interaction. At that time K3 plays the role of the atomic bound state. Next, consider 4 = n, i.e., X is negative real, and V, is repulsive. When 1 h 1varies from 0 to co, the poles K~ and K~ move towards each other a bit (slightly less binding, reasonable, the repulsion is felt only in a small part of the atomic volume). They reach then the positions ~3~ and ~3~. The poles ~3 and K~ move always from -fl in opposite (almost) vertical directions towards fico, as shown in Fig. 6. In Fig. 6 we have also sketched trajectories for several values of I$ satisfying 0 < #J < rr. Because A(K) is known explicitly they can be easily calculated as the lines of steepest ascent of the modular landscape ofA( Begin (I X 1 = 0) and end (I X 1 = cc) points are fixed, independent of #J. Also the directions in which the poles leave their original positions are easily found, K~ leaves the real axis under an angle 4, K~ has angle $ $ n, ~3 has angle $12, and K~ in the opposite direction (saddle). If 1x 1 approaches co two poles approach ~3~ and ~3~ under an angle -4 with the positive and negative real axis, respectively, the other two disappear at infinity under opposite directions, which have an angle $/2 with the real axis. It turns out that in the present example we have a saddle point in the physical region. For small positive values of 4 (attraction plus small absorption) the trajectories deviate little from the trajectories for the case of X real and positive. The pole ~3 always travels to K3*, and K~ travels to infinity, both trajectories have been pushed upwards slightly. When we increase 4 the saddle point KS makes itself felt. For the coupling parameter h Crit = A the two trajectories of K~ and ~3 meet head on. Denoting +,rit = arg hcrit we observe the phenomenon of tail swapping (cf. Section 3) when $ m &it. For 4 > &it ~3 no longer travels to ~3@, but towards infinity, and K~ travels towards K3 m instead. Upon increasing 4 further to rr no further saddle points are met. The trajectories vary as sketched in Fig. 6, thus reaching the position for a repulsive potential (I$ = n). We have treated the present example in some detail, because it clearly exhibits all basic features of more intricate situations. In fact, in such situations there may be many more saddle points in the physical region. Consequently, if the basic mechanism is not properly understood, those cases may easily lead to false interpretations of the movement of poles. EXAMPLE 4. The last example is the attractive Coulomb potential -2s/r, which clearly is of great physical interest. It gives an infinite number of bound states located at the positive real K axis at K = s/n, n = 1, 2,..., with an accumulation point at K = 0. The introduction of the separable term (2.1) with complex-valued coupling parameter h causes a shift of the bound-state poles. The picture of moving poles in the complex plane associated with variation of X is again very illustrative. In fact, for real X these movements of the bound-state poles have been studied before [3, II].

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439

SCATTERING

For h > 0 there will be at most one additional bound state, cf. Section 2, and Ref. [3]. When, starting at X = 0, X is increased, h -+ $00, the bound-state poles starting at s/(n + 1) all move to the right to a point K~=, II = 1, 2,... . This point lies left of the next pure Coulomb pole. The small parameter determining the final distance s/n - K,” will be the ratio of a typical range of the separable force (say, p-l) and a typical size of the Coulomb system (say, the Bohr radius s-l). The ground state, starting at K = s, moves to + 00, behaving as a typical short-range interaction bound state. For real negative h, when X -+ -cc, all bound-state poles travel the small distance to the adjacent IC,~, the additional repulsion reduces the binding. For the form factor (4.5) the function A(K) can again be calculated explicitly [1 I], &)

= @+ B24” (1 -~)/2~l(1,-~,2-f;(~)2).

(4.9)

Now consider complex values of h. For 4 = 0 the Coulombic poles K, move to the right, for 4 = rr to the left, when j h j is increased from zero. What happens for r ? For small values of 1X / the Coulombic poles leave the axis under an o<+< angle 4. For large values of I h ) the poles arrive back at the real axis (at the points K,“) under an angle rr - 4. The trajectories of the poles for intermediate values of 1X 1 are strongly influenced by saddlepoints. The modular landscape of (1(K) of (4.9) has an infinite number of saddle points. Presumably almost all of these lie in (or at least close to) the physical half-plane. There is a close analogy with the previous example, although in that case only one saddle point was situated in the first quadrant of the K plane. There, for large (b (for #erit < sb < z), the “strong” pole starting at -/3 moved into the physical half-plane along a path above the saddle point. For small +(O < 4 < &it) the pole moved towards the “Coulombic” region, and the pole originally representing the Coulombic ground state moved towards infinity. In the present example we expect that for large 4 (for rr 2 4 > &rit,m) again a “strong” pole will drift towards infinity along a path above the Coulombic region and above al1 saddle points, when / X j is increased to co. It is likely that for smailer values of + the nth Coulomb pole will move towards infinity, if we choose a particular fixed value of 4 with * > #crit,n > #J > $crit,n-l , 12= 2, 3 ,... . For small values of 9 (&it,r > r$ 3 0) the Coulomb ground state starting at s will move towards infinity. If we wish to select a ray in the complex h plane such that the ith Coulomb pole moves towards infinity, we merely have to choose 4 E arg ;Z between two adjacent values 4erit, so that the path C, passes between the matching two saddle points. All corresponding other paths C: starting at s/n, n < i, will reach the (left) adjacent K,” (less binding energy). 411 corresponding other paths C: starting at s/n with it > i will move on the average to the right to K;-~ (more binding energy). 5. MORE GENERAL PERTURBATIONS

Vs

In Section 2 we studied the effects of the addition of a rank one separable interaction (2.1) to a hermitian Hamiltonian H. In this section we make the extension to rank N .595/131/2-I4

440

KOK

AND

VAN

HAERINGEN

separable additions, vs = - 5 At IgtXgt

I.

(5.1)

i-1

Each of the coupling parameters X8 can be taken complex. Throughout we follow Section 2 closely. The total potential now is

Wl , A2,...,hv)= v - d-15 I$2)At<&I.

this section

(5.2)

The Green operator is given by (cf. Eq. (2.8))

W4 ,..., hv) = G - $, G I gt> r&j I G, t,j=l

(5.3)

where the matrix elements Tii follow from

(+)ij = (Wtj + .

(5.4)

By sandwiching Eq. (5.3) between ( g, I and I gm) one can write (5.5) where the symbol h denotes matrices in an N dimensional space spanned by the vectors I gJ. Here X as a subscript means the sz of all Xi , i = l,..., N, whereas fi is a diagonal matrix with elements Ai 6ij . Note that h-l = A--l. In analogy with Eq. (2.9) one derives

G,l = e-1 + A. It follows from Eq. (5.3) that in order to find the poles of G(& ,..., X,) one has to investigate the poles of G and the poles of 7ij . From now on we restrict ourselves to N = 2. We introduce the notation

gtj E < gi I G I gj>Furthermore,

let D be the determinant D = Wet 4-l

of the 2

= K1 + g&G1

x

2 matrix 4,

(5.7) so

+ &I - g18gsl .

(5.8)

Note that for E real negative g,, = gjd and Tij = rji . According to Eq. (2.6) G has simple poles at E = -K,,~. We see from Eq. (5.3) that any pole E of G(X, , X3 is (i) (ii)

either equal to a zero of D or equal to an eigenvalue -

K,~

of H.

COMPLEX POTENTIAL

SCATTERING

441

We have proved elsewhere that the latter possibility can in general be excluded [3]. Therefore the vanishing of the right-hand member of Eq. (5.8) is the natural generalization of the basic relation (2.12) of this paper. 6. DISCUSSION

In the previous sections we have considered rather different examples of hermitian Hamiltonians H = H,, + V, viz., a vanishing V, a rank one separable V, and the local Coulomb potential I’. In the first three examples of Section 4 the function A(K) is an analytic function of K in the whole K plane, except for isolated poles. In the Coulomb case A(K) has an infinite number Of p&S Knm at the positive K axis, corresponding to zeros of the hypergeometric function in the denominator of the right-hand side of Eq. (4.9). The other singularities of n(K) are due to the branch-type singularities of this zFl , which has one branch cut from 0 to -/3, and another from -p to -co. The singularity structure ofA( can be found from the representation (2.17). Analyticity properties of G(E) have been studied extensively for wide classes of potentials. We shall not study the analyticity properties of II(K) and of G(E) here. Instead we refer to the literature [ 1, 91. The addition of a separable interaction (2.1), with h complex, to a given Hamiltonian leads to shifts of the bound-state poles (and of the resonance poles, as is shown in what follows). These shifts are determined by the function (1(K), defined through Eq. (2.15). Sometimes, as a function of one of the parameters / h 1, arg h, Re A/lm X, Re X, or Im h, the trajectories described by the poles K show an unexpected behavior. This behavior is associated with the occurrence of saddle points in the modular landscape of A(K). In model calculations of physical systems, which can expected to be described well by a complex interaction (nucleon-antinucleon systems), such a phenomenon has been found “empirically” in the papers of Ref. [5]. There it is observed that the pole trajectories as a function of / h 1can be greatly different if a small change is made in the ratio Im h/Re h around certain critical values. From the discussion in the previous sections it is clear that one may expect critical values of h associated with the saddle points in the K plane. It should be stressed that h itself has a critical value rather than arg A. This value, herit, is in general complex. (Indeed, one expects a critical value of I h 1, when one studies pole trajectories as a function of q!~= arg h, at ; &it j. A round this value the trajectories will exhibit large global changes). Also Kudryavtsev [12] observes a phenomenon of criticality, in a two-channel case with separable coupling. Upon increasing coupling certain resonances move upwards, whereas others move downwards. In order to get some more insight in cases with decaying bound states or resonances we reconsider Example 3 of Section 4. We take p’ = 0.25 (“long range”) and /3 = 1.0 (“short range”). The ratio short range/long range has not been chosen as small as, for example, in exotic atoms occurring in nature (K mesic atoms, pp, etc.), where it can be less than a few percent. The reason is that the figures in the following would become unclear. As the first case we consider

442

KOK

AND

VAN

HAERINGEN

long-range attraction, h’ = + 1.96. This attraction by itself is strong enough to support a bound state, located at K = 0.1. The level lines (broken), and the steepest ascent lines (solid) in the complex k plane of the modular landscape of A(K) are given in Fig. 7. (To obtain the K plane one rotates the journal by -&r. From here on we shall work in the k plane, k = in). Along the solid lines 1X 1varies from 0 to co, along the broken lines arg h varies. Only solid lines corresponding to absorptive or real potentials (Im h 2 0) are shown. Figure 8 shows, for the same case, the modular landscape of exp A(K). Here along solid lines Re )t varies, whereas along the broken lines Im h varies. Again only solid lines corresponding to absorptive or real potentials are shown. Both in Fig. 7 and in Fig. 8 the saddles are clearly seen. They occur at the same points in the two figures for the following values &it , K

=

K

=

K

=

-0.690 -0.332

K

=

0.0114

K=O[)

--I

&

0.284i

for for for for for

&it &it &it X,,h &it

= = = = =

0.0, 0.0457, 1.382, 0.842 & 1.1 lOi, 00,

Note that the Coulombic pole (i.e., the pole in the neighborhood of the origin) describes a counterclockwise more or less circular path under variation of ,Re h from

FIG. 7. Trajectories landscape of A. Along increasing / X I. Along Parameters: X’ = 1.96,

in the k plane, given as level contours, and steepest ascent lines of the modular solid lines \ X 1 varies, arg X is fixed. The arrows indicate the direction of broken lines arg h varies, and 1 h [ is fixed. All solid lines have Im h > 0. /3 = 1, 8 = i.

COMPLEX POTENTIAL Im

443

SCATTERING

k

/’

L -06

0

06

FIG. 8. Trajectories in the k plane. Among solid lines Re h varies, along broken lines Im h varies. The arrows indicate the direction of increasing Re h. Same parameters as in Fig. 7.

- 03 to + co, provided that Im h is larger than Im Xcrn for the nearby saddle (i.e., for Im h > I.110 only). Our case serves to illustrate two phenomena observed by Krell, and subsequently simulated and explained by Koch et al. [7]. They consider a large range attraction, and a complex short-range potential. (i) For large values of Im h the optical potential was found to be repulsive, i.e., it decreases the binding energy of the Coulombic level. In our case this is clearly seen, too. In fact for any value of X for which j h / is large there is a Coulombic pole just below (in the k plane) the original Coulombic pole corresponding to X = 0. (On the scale of Fig. 7 this shift is rather large to enhance visibility of the effect. As a consequence the “Coulombic” level even becomes unbound in this case. This is merely due to the relatively large value of ,&‘I/3 = g.) Note the existence of another pole in the k plane near infinity in the direction of the left upper corner of Fig. 7. The figure clearly shows that it may be that the original Coulombic pole has travelled via a connected path to that.point, whereas the pole described by Krell, Koch et al. has come via a connected path from the unphysical half-plane (from k = -i/3, if we start at h = 0). (ii) For intermediate values of Im X the calculated energies and widths each exhibit oscillations (about +r out of phase with each other) as Re ;\ is increased. For this phenomenon

one should watch the movement

of the Coulombic

pole in

KOK AND VAN HAEXINGEN 0.039

0.038

I.037

I -0.ocl2

I - 0.001

J

FIG. 9. Pole trajectory in the E plane, corresponding to Fig. 1 of Koch et al., Ref. [7], corresponding to varying (indicated) real part of the coupling constant.

Fig. 8. For Im h not too small (otherwise the pole wanders off to infinity) and not too large (otherwise the previous phenomenon dominates) we note a circular path, going counterclockwise. We have redrawn Fig. 1 of Koch et al. as our Fig. 9, in a Re E vs Im E plot. We note seven more or less circular movements of the pole, for the fixed intermediate value of Im A, as a function of Re A. The occurrence of seven oscillations is explained by the fact that here a local complex potential is considered. Indeed, from experience in few-body physics and other parts of nuclear physics we know that a convenient and good representation for a short range local potential is a finite rank separable potential. If the local potential has seven bound states one needs at least seven separable terms. In fact the crosses in Fig. 9 indicate the values of Re X where the real short-range potential permits an additional bound state (“a new inner state” [7]). Each cross turns out to mark one circular movement. On the other hand it also marks a new bound-state wave function, and the need of an extra separable term in the expansion of the local potential. Indeed, the connection between the bound-state wave functions and the form factors is well understood. The reason for the clockwise motion

COMPLEX POTENTIAL

SCATTERING

445

in Fig. 9 (rather than the counterclockwise motion in Fig. 8) is trivial and lies in the fact that Koch et al. consider positive-energy bound states in a box-type potential. The general trend of the pole movement (i.e., higher energy, less binding, and decreasing width) can be understood from the general rules governing the motion of poles as derived for the separable interaction. The fact that the effect occurs for intermediate values of Im h agrees with the observation that a necessary condition for the circular path for the separable interaction is Im h > Tm &it (in Fig. 8 this requirement is Im X > 1.1 lOi). In numerical work we have observed that the value Im hcrit decreases with increasing /3. Recently the phenomenon of reconstruction of spectra, discussed originally by Zel’dovich [6], again received attention [13, 141. When a strong short-range potential is added to a long-range attractive potential the bound states of the latter are shifted. When the depth of the short-range potential is slowly increased, the long-range (Coulombic) levels quite suddenly move quickly downwards towards the next lower level, so that the original long-range spectrum is approximately restored. The sudden jump occurs when the short-range potential by itself becomes strong enough to support an additional bound state. If we take the short-range potential separable with strength h (cf. Eq. (2.1)) such a jump happens only once, whereas for local potentials it happens more often. For complex h the movement of poles again is very illustrative. For arg h = 0 the separable term first introduces a virtual level which then, upon increasing A, takes the place of the highest Coulombic level. Roughly speaking all Coulombic levels cascade down one position, with the exception of the ground state, which rapidly

FIG. 10. Trajectories in the k plane. / X / varies along solid lines, arg h varies along broken lines. The arrows indicate the direction of increasing 1h I. Parameters: h’ = -5, j3 = 1, j3’ = 4.

446

KOK AND VAN HAERINGEN

becomes bound more strongly, as discussed in Example 4 of Section 4. If Im h becomes positive the Coulombic levels may start to describe circular paths. Yet we expect spectrum reconstruction, here, too. For local short-range potentials we expect the total effect to be similar to the successive effect of the addition of as many separable terms as the short-range potential has bound states. Myhrer and co-workers in several papers [8] describe the influence of absorption on possible nucleon-antinucleon resonances and bound states. In order to allow for resonances we now choose in the third example of Section 4 a repulsive long-range potential (we take A’ = -5, and the same value of p and /3’ as before). The modular landscapes of li and of exp A are shown in Figs. 10 and 11, respectively. Solid and broken lines have the same meaning as in Figs. 7 and 8. The region covered by the full lines corresponds to absorptive (or real) potentials (Im X b 0). Close to the positive k axis (physical region) there is an ellipse-shaped region. Increasing Im h from 0 to positive values corresponds to moving inwards from its boundary in Fig. 11 along broken lines. Indeed, as shown by Myhrer and Thomas, and by Myhrer and Gersten, resonance poles move generally downwards, i.e., resonances become broader. Only inside the lower part of the ellipse resonance poles move upwards in the k plane. Consequently very broad resonances may become a bit less broad upon increasing absorption, depending on Re A. The movement of the bound-state poles follows trajectories (broken lines) perpendicular to the full lines in Fig. 11, when Im X is increased. They start at the positive imaginary k axis. Of primary importance here is the saddle just left of the origin.

-0.6

0

0.6

FIG. 11. Trajectories in the k plane. Re X increases along solid lines in the direction indicated by the arrows, Irn X varies along broken lines. Same parameters as in Fig. 10.

COMPLEX

POTENTIAL

447

SCATTERING

So far we have considered cases with Im h = 0 and Im h > 0, only. The latter case corresponds, roughly speaking, to absorption. The case Im h < 0 thus corresponds to creation. For hermitian Hamiltonians reversing the sign of Im X merely induces a reflection of all pole trajectories with respect to the imaginary k axis. It is interesting to study in the present example the regions in the fourth quadrant of the complex k plane where poles can occur for absorptive values of h. For various values of h’ this is shown schematically in Fig. 12. If the long-range part is attractive (Fig. 12a) there are no poles close to the physical region. For repulsive long-range potentials for small fixed negative /\’ first a situation as in Fig. 12~ develops. The contours separating shaded (absorptive) and white (“creative”) regions correspond to real values of h. Here the resonance can turn into a virtual state at a small interval below the origin, and finally into a bound state at the positive imaginary k axis. This situation is analogous to the case of local potentials where a resonance caused by a long-range barrier with increasing short-range attraction sinks into the well of bound states (cf. Fig. 13). For more negative h’ we have a situation as given in Fig. 12d. Here, with increasing h the path of the bound-state pole is no longer connected with the resonance region. In fact the bound-state pole travels along the negative imaginary k axis as a virtual-state pole, before entering the physical half of the k plane. This virtual-state pole has not touched the shaded resonance region. We note in passing that the dissociation of the drop-like region in the k plane is associated with a singularity in the parameter x’, which occurs at a real value. Apparently the visualization of Fig. 13 no longer is strictly valid. Instead we have the development according to Fig. 14, where we have sketched the energy levels with increasing shortrange attraction. In the same figure the corresponding levels are indicated when a small amount of absorption is added. The most important difference between the location of the levels in Fig. 13 and that in Fig. 14 concerns the size of the gap between the lowest possible resonance and the threshold E == 0. In addition in Fig. 14 the resonance in case IV has become more diffuse than that in case III. In contrast.

FIG.

XL196

x':O

!ill

lb)

f;.

196

ICI

h’= -5 Idi

12. Schematic picture of regions in the k-plane (shaded) where poles with absorptive

potentials can occur. Parameters: p = 1, ,3’ = b. Values A’ are (a) I .96, (b) 0, Cc) -- 1.96, and (d) -- 5. Saddle points are indicated by a cross.

448

KOK AND VAN HAJZRINGEN E.Vlrll

FIG. 13. Sketch of resonance and bound-state levels for a local potential with large-range repulsion and small-range attraction of varying strength.

in Fig. 13 theiwidth of the resonance decreases monotonically. Both effects probably are intimately connected with the nonlocal character of the interaction. The saddle points in the k plane have been indicated by crosses in Fig. 12. Note that in all these cases there is one additional saddle at infinity.

7. CONCLUDING

REMARKS

In this paper we have studied the effect of complex separable potential perturbations on the negative-energy eigenvalues of a given Hamiltonian, on resonance and virtualstate poles, and on so-called left-hand singularities. The relation between the complex coupling parameter h and the location of the poles is of the type X = A(K). The modular landscape of A(K) determines pole trajectories associated with varying A. Important features of this landscape are its zeros, its poles, and its saddle points. These saddle points are linked to branch-type singularities in the complex h plane. Methods in

FIG. 14. Resonance and bound-state levels for various strengths of the attractive separable potential. Both cases with and without absorption are sketched.

COMPLEX

POTENTIAL

449

SCATTERING

physics, based on the (analytic) continuation of certain physical properties in the coupling constant should be used with great care only. It should be stressed that saddle points of the above type can and do occur in the physical half of the k (and K) plane. In this context it is important to note that for angular momentum larger than zero in the case of short-range attraction and intermediate-range repulsion always a saddle occurs at K = 0. The corresponding (real) branch-point singularities in the coupling parameter, which are kinematical in origin, have been taken care of explicitly, in the work by Kukulin, Krasnopol’skij, and Miselkhi [15], who do use the promising device of continuation in the coupling parameter. It seems more difficult to take care of the branch singularities in X, associated with the saddles discussed in this paper which are much more dynamical in origin, in such a continuation procedure. Yet this will be necessary for excursions into the complex X plane. Summarizing we conclude that the behavior of poles as a function of the coupling parameter can be at first sight unexpected and surprising due to the occurrence of saddle points. In Section 6 we have discussed various phenomena described in the literature. Various extensions of the present work are of interest. First, what is the effect of complex local potential perturbations ? This point has been touched upon in the discussion of the results of the work in Ref. [7]. Studies in various fields of physics have revealed that local short-range potentials are well approximated by finite (low) rank separable potentials. (Note that the required rank depends on the strength of the potential). It seems reasonable to expect that such a local perturbation has (in good approximation) the same effect as its rank N separable approximation. In Section 5 we discussed a rank two potential with strength parameters h, and X, . As long as one of these is real we may include the corresponding term in the Hamiltonian H = H, + V by simply adding it to V. We are then back at the N =- I case. When both h, and X, are complex, we may fix one of the values, say h, , and then study the trajectories C, corresponding to variations of h, = h in the complex X plane. In effect we then have the N = 1 case, too, however, with a nonhermitian H. Here we may have to invoke known techniques to deal with such nonhermitian Hamiltonians [ 1, 91. As long as the domain of anaiyticity Of (1(K) in the K plane exists, this seems to be no principal difficulty. Some care is needed in this case because the trajectories C, are no longer symmetric with respect to the real K axis under sign change of Im X. We also can study the rank two potentials with h, = h, = h taken complex. ln this case we may use Eq. (5.8) to derive a relation analogous to Eq. (3.1) where now 2/l(K)-1

=

--g,,

-

g22

-

((811

-

&2Y

+

4g12g2,P2.

(7.1)

It is of great interest to study this case in more detail, and to extend this investigation to the more-channel case with different threshold energy values. Important work in this direction has been done by Karlsson and Kerbikov [16]. We hope to be able to further investigate this problem in the near future. It is reasonable to expect qualita-

450

KOK AND VAN HAERINGEN

tively the same phenomena as described in this paper. Indeed Koch et al. [I71 for a two-channel case report findings similar to the one-channel case. Furthermore, many of the references quoted in the previous sections do consider two-channel cases.

ACKNOWLEDGMENT Much of this work is fruit of stimulating discussions with Professor Yu. A. Simonov. This work is part of the Project “The Coulomb Potential in Quantum Mechanics and Related Topics.”

REFERENCES 1. See for example, V. DE ALFARO AND T. REGGE, “Potential Scattering,” North-Holland, Amsterdam, 1965; R. G. NEWTON, “Scattering Theory of Waves and Particles,” McGraw-Hill, New York, 1966; H. M. NIJ~SENZVEIG, “Causality and Dispersion Relations,” Academic Press, New York, 1972; A. G. SITENKO, “Lectures in Scattering Theory,” Pergamon, Oxford, 1971; J. R. TAYLOR, “Scattering Theory,” Wiley, New York, 1972. 2. K. W. MCVOY, Nucl Phys. A 115 (1968), 481, 495; C. J. GOEBEL AND K. W. MCVOY, I&cl. Phys. A 115 (1968), 504; K. W. McVou, in “Fundamentals in Nuclear Theory” (A. de-Shalit and C. Villi, Eds.), p. 499, Intern. At. Energy Agency, Vienna, 1967. 3. H. VAN HAERINGEN AND L. P. KOK, J. Math. Phys., in press. 4. H. A. LAUWERIER, “Asymptotic Expansions,” Math. Centre Tracts 13, Amsterdam, 1966. 5. M. I. POL~KARPOV AND Yu. A. SIMONOV, Preprint ITEP-162, Moscow, 1978; J. C. H. VAN DOREMALEN, M. VAN DER VELDE, AND Yu. A. SIMONOV, Phys. Letr. B 87 (1979), 315; J. C. H. VAN DOREMALEN, Yu. A. SIMONOV, AND M. VAN DER VELDE, Nut!. Phys. A 340 (1980), 317; M. VAN DER VELDE, Ph. D. thesis, Free University, Amsterdam, 1980. 6. YA. B. ZEL’WVICH, Sou. Phys. Solid State 1 (1959), 1637. 7. M. KRELL, Phys. Reu. Lett. 26 (1971), 584; J. H. KOCH, M. M. STERNHEIM, AND J. F. WALKER, Phys. Rev. L&t. 26 (1971), 1465. 8. F. MYHRER AND A. W. THOMAS, Phys. L&t. B 64 (1976), 59; F. MYHRER AND A. GERSTEN, Nuovo Cimento A 37 (1976), 21. 9. T. K. BAILEY AND W. C. SCHIEVE, Nuovo Cimento A 47 (1978), 231. 10. V. F. KHARCHENKO, N. M. PETROV, AND S. A. ST~ROZHENKO, Nucl. Phys. A 106 (1968), 464. 11. H. VAN HAERINGEN, C. V. M. VAN DER MEE, AND R. VAN WAGENINGEN, J. Math. Phys. 18 (1977), 941. 12. A. E. KUDRYAV~EV AND R. T. TYAPAEV, Yud. Fiz. 30 (1979), 1609. 13. A. I. NIKISHOV AND V. I. RITUS, Zh. Eksp. Teor. Fiz. 52 (1967), 223 [Son Phys. JETP 25 (1967), 1451. 14. A. E. KUDRYAVTSEV, V. E. MARKUSH~N, AND I. S. SH~SIRO, Zh. Eksp. Teor. Fiz. 74 (1978), 432 [Sov. Phys. JETP 47 (1978), 2251. 15. V. M. KRASNOPOL’SKY AND V. I. KUKULIN, Phys. Lett. A 69 (1978), 251; V. I. KUKULIN, V. M. KRASNOPOL’SKI~, AND M. MI~ELKHI, Yad. Fiz. 29 (1979), 818 [Sou. J. Nucl. Phys. 29 (1979), 4211. 16. B. R. KARLB~N AND B. KERBIKOV, Nucl. Phys. B 141 (1978), 241. 17. J. H. KOCH, M. M. STERNHEIM, AND J. F. WALKER, Phys. Rev. C 5 (1972), 381.