Schematization of scattering amplitude and wave function for potential resonance using Weinberg's quasiparticles

2.C

I

Nuclear Physics A138 (1969) 442--456; @) North-Holland Publishing Co., Amsterdam

i

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

SCHEMATIZATION FOR

POTENTIAL

OF SCATTERING RESONANCE

AMPLITUDE

USING

AND WAVE FUNCTION

WEINBERG'S

QUASIPARTICLES

R. ItUBY

Department of Theoretical Physics, The Chadwick Laboratory, University of Liverpool, U.K. Received 15 August 1969 Abstract: The problem is to describe the resonant scattering of a partial-wave of angular momentum I (non zero) in an attractive potential V(r), by a schematic approximation which displays energy-dependence of Brcit-Wigner type and which possesses other specified desirable features. The idea is to work at real, positive energies E (without analytic continuation), and to produce a good approximation not only for the scattering amplitude but also for the scattering wavefunction g.(E, r), the latter to depend smoothly on r (no joining radius), to have a "nearly bound" shape within the potential, and to have a proper long-distance tail. This is achieved through Weinberg's "quasiparticlc" decomposition of the equations, which makes use of the cigenfunctions and cigenvalues of the Lippmann-Schwinger kernel. Two alternative schemes arc used: Weinbcrg's original T-matrix one, and a modified K-matrix one, the latter being more accurate. For the scattering function S~(E) formulae are obtained which exhibit a background and a resonance contribution, the formula in the K-scheme being structurally very close to that of R-matrix theory. The width Fo in the Tscheme is z-ct <~eo(Z~o) I VIZ(Eo)>[ z whcre ~po(Eo, r) is the Weinberg quasiparticlc wave function (normalized) and Z(Eo, r) is a free wave function at the resonance energy Eo. The scattering wave function ~p(E, r) contains a resonant term proportional to ~0o(E,r), bearing art amplitude showing Breit-Wigner energy dependence. The interpretation of the Ibrmulae is that the resonance mechanism is associated with a discrete resonance state, viz. the quasiparticlc state ~o(E, r), which the particle from the frec wave makes a transition into, and out of again.

1. Introduction A l t h o u g h t h e r e s o n a n o z s m o s t typical o f n u c l e a r r e a c t i o n s arise f r o m the c o m p l i c a t e d states o f m a n y particles ( c o m p o u n d - n u c l e u s states), a r e s o n a n c e can also o c c u r in the s c a t t e r i n g o f a single p a r t i c l e by a n a t t r a c t i v e p o t e n t i a l V(r) in a p a r t i a l w a v e o f a n g u l a r m a m e n t u m l g r e a t e r t h a n zero, the p a r t i c l e b e i n g c o n f i n e d w i t h i n the well by the c e n t r i f u g a l barrier. S u c h s i n g l e - p a r t i c l e r e s o n a n c e s m a y be studied for t w o r e a s o n s . T h e first m e t h o d o l o g i c a l r e a s o n is t h a t it o u g h t to be p o s s i b l e to e x e m p l i f y any syst e m a t i c t h e o r y o f r e s o n a n c e s in the basic p o t e n t i a l m o d e l . S e c o n d l y , a s i n g l e - p a r t i c l e r e s o n a n c e m a y a c t u a l l y f o r m o n e c o m p o n e n t (or c o n f i g u r a t i o n in s h e l l - m o d e l t e r m s ) in a m o r e c o m p l i c a t e d r e s o n a n c e o f m a n y n u c l e o n s . T h e i n c o r p o r a t i o n o f a singlep a r t i c l e r e s o n a n c e in a m o r e c o m p l i c a t e d r e s o n a n c e will be the subject o f a s e p a r a t e paper. Since the S c h r d d i n g e r e q u a t i o n for a p a r t i a l w a v e l in a p o t e n t i a l V ( r ) at e n e r g y E c a n r e a d i l y be s o l v e d b y n u m e r i c a l m e t h o d s , o n e m i g h t strictly s p e a k i n g d o n o m o r e 442

SCATTERING AMPLITUDE

443

than to trace out what happens in any particular potential as the energy passes through a resonance. However, there are advantages in seeking a schematic theory which exhibits such simple, general features as the occurrence of a Breit-Wigner factor (E_Er+½_iF)-t in the scattering amplitude, together with a simple formula for the width F. Numerous powerful systematic theories have been devised up to the present, but they have their shortcomings, as pointed out in a critical review 1). This stands out particularly if we seek a simple formula for the wave function near resonance. For example, in Wigner's R-matrix theory 2) the arbitrary treatment employed at the joining radius a tends to give an approximate wave function an unrealistic shape with discontinuity at a. Again, it is often held that the most natural definition of a resonance energy is that used in the theory of Humblet and Rosenfeld 3), viz. as the complex energy at which the scattering function S has a pole. However, apart from cther difficulties, this means that the wave function most directly associated with the resonance is the " G a m o w " wave function at the pole, which increases exponentially at large distances. The principal objective of the present paper has been to find a schematic theory yielding a satisfying approximation for the scattering wave function around resonance, as well as the scattering amplitude. This involves identifying some discrete resonant state whose wave function inside the well resembles that of a bound state, and Maich shall appear in the formula for the physical scattering wave function with an amplitude bearing the Breit-Wigner factor (E-Er+~iF )-1. The scattering wave function must nevertheless continue out smoothly through the centrifugal barrier, with a realistic tail at large distances. The solution proposed is to identify' the discrete, resonant state with a quasipartic!e state of Weinberg's theory 4) or the modification of this theory to standing waves 5, 6). Most schematic theories of resonance lead to a picture of the process according to which the particle makes a transition out of its incident, free wave into a discrete, resonant state inside the field of force, and then back again. Thus in the R-matrix theory a) the role of this discrete state is played by an inside eigenfunction )¢5. of the total Hamiltonian H; and in a many-channel theory by an eigenfunction of the projected part QHQ in a subspace of closed-channel states, [e.g. Feshbach 7)]. The same role is played in the present theory by the quasiparticle state, thus assimilating the case of potential resonance to other resonance mechanisms. The formalism of Weinberg's theory for scattering as here developed is sufficiently complete that, although crude approximations have to be made to obtain a very simple description of a resonance, any greater degree of accuracy can be achieved by the retention of correction terms. As explained in refs. 5, 6), Weinberg's theory can be developed in two forms: either the original theory in terms of outgoing eigenfunctions and the T-matrix, or a modified theory in terms of standing waves and the K-matrix. The former is simpler and more direct, and is therefore presented filst here. However, because of the complex numbers involved in the former scheme, one has to make frequent approximations which are difficult to adjudge, in order to obtain practical formulae. With the second

444

R. HUUY

scheme's basically real numbers these uncertainties do not arise, and so we present also this alternative scheme, as a check and a correction to the former.

2. The quasi-particle formulation of scattering in a potential In ref. 5) a r6sum6 was given of the theory of scattering in a potential V(r) based on either the T-operator or the K-operator, together with a formulation of the solution in either scheme in terms of a set of quasiparticle states as devised by Weinberg [rcf. 4)]. The purpose of Weinberg was to permit a perturbative solution by means of a "quasi-Born" series. While following ref. 5) for most of the notation and detailed theory, we recapitulate here some of the essentials. The free particle has a Hamiltonian Ho = - V 2,

(2.1)

to which is added a central potential * V(r), which will be assumed everywhere attractive (negative) and otherwise well-behaved [eq. (2.2) of rcf. s)]. We study scattering at energy E = k'-

(2.2)

in a partial wave of angular momentum l which will later be chosen non-zero. The incident wave is thus one of the eigenfunctions of Ho, which we choose normalized in the energy-scale: z(E, r) - z(E, r) itYz,m(O, ~),

(2.3)

r

with

z(E, r) = \./½rS,+~(kr)

(as

,. --,

-

(2.4)

sin

so that = a(e-

e').

(2.5)

If we now define the Lippmann-Schwinger kernel I(E+i~) by 1

I(E + ie) = . V, E+ i~-H o

(2.6)

and its resolvent F(E+i~) by 1 +F(E+

i~) --

1

a-I(E+i )

,

(2.7)

It would be possible to extract a portion of the potential for inclusion in Ho, whose eigenfunctions would then be "distorted waves".

SCATTERING AMPLITUDE

445

the exact scattering wavefunction at energy E is

~(E, r) = 0(E, r) i'Y,,m(0, 4,) r

= [1 + v ( ~ + i ~ ) ] x ( ~ , r).

(2.8)

The reaction operator T is defined by

T(E+ ig) = V[1 +F(E+ ie)],

(2.9)

and then its on-shell matrix element Tt(E), defined by Tz(E) = (z(E)IT(E+ i~)[z(E)),

(2.1o)

yields the scattering function St(E) by

St(E) = 1-2ziTz(e).

(2.11)

The asymptotic form of the scattering wave function is related to Tt(E) by 0(E, r) ,,,

[sin (kr--~.nl) -nT~(E) exp i(kr-½nl)].

(2.12)

In the alternative standing-wave scheme the kernel I(E+ ie) is replaced by

J(E) = P - __1.... V, E-H o

(2.13)

where P indicates the princiPal part. The corresponding resolvent G(E) is defined by

l + G(E) -

1

1-J(E)

(2.14)

An exact, standing-wave eigenfunction of H is then

O(E, r) - (o(E, r) ityt,,,(O, q5) Y

= [1 + G(E)]z(~, r),

(2.15)

with ~b(E, r) a real function. The reactance operator K is defined by K ( E ) = V[1 + o ( e ) ] ,

(2.16)

and its on-shell matrix element Kt(E) is defined by

K,(E) = (z(E)IK(E)I)~(E)),

(2.17)

which is real. We can obtain Tt(E ) from KI(E) by

T~(E) -

Kt(E) 1 + ircKt(E)'

(2.18)

446

R. HCBY

and the scattering wave function ~(E, r) from the standing wave function ~b(E, r) by ~/,(E, r) =

q$(E,r)

1 + iTzK;(E)"

(2.19)

The basic problem is the determination of the operator F(E+ ie) or G(E). In Weinberg's method one subtracts off from V a number of separable terms, writing V = V'+ Vo,

(2.20)

where N

VIs)(5[V ,

17o = ~ s=l

(2.21)

<51Vls)

the vectors Is) and = 10s(E)>,

(51 -- (~s(E)l,

(2.22)

where g,,(E, r) is an eigenfunction of I(E+ie) with complex eigenvalue ~,{,E), and ~,(E, r) is an "adjoint" function. The single quasiparticle state which is to be used will be written:

~o(E, r)

-

u(E, r)

i'Yz,=(0, q~),

(2.23)

1"

~o(E, r) - u*(E, ,') iiY,,,,(O, cb),

(2.24)

I"

and the eigenvalue will be denoted ~(E). In the standing-wave or K-scheme the ideal quasiparticle state is: Is) = Iq$,(E)), where

rk~(E,r) is an

eigenfunction of

J(E)

(51 = (q$,(E)l,

(2.25)

with real eigenvalue p,(E). We shall write

SCATTERING AMPLITUDE

447

for the single quasiparticle used: 40(tr, r) = v(e, r) i'r,,,,(O, O),

(2.26)

r

v(E, r) being a real function, and the eigenvalue will be denoted [3(E). All the formulae of this section are independent of any normalization chosen for these various eigenfunctions. The residual potential V' now has different meanings in the two schemes. In the T-scheme it is

V' = V - VI~'°(E))<(~°(E)IV

(2.27a)


v ' = v - vloo(E))<4o(E)lv

(2.27b)

<,~o(E)lVl~Po(E)>

In terms of these quantities, the exact solution for F or G [as derived in ref. 5) eqs. (4.16a and b)] is

F(E + ie) = F'(E+ ie) +

~(E)I¢o(E))((Jo(E)IV

(2.28a)

[1 -c~(E)]<~o(E)l V[Oo(E)>' #(E)lOo(~))(0o(e)l v G(E) = G'(E) + [1 -- fl(E)](q~o(E)[ Vkbo(E)) "

(2.28b)

The corresponding solutions for T and K are

T(E+ie) = T'(E+ie)+

V[Ip°(E))(~°(E)[V [1 - c,(e)](~o(e)l

(2.29a)

VI~,o(E);, '

Vl4Jo(E))(epo(E)lv K(E) = K'(E) + [1 -- fl(E)](qSo(E)] VlqSo(E)) "

(2.29b)

In the first-order quasi-Born approximation, both T' in (2.29a) and K' in (2.29b) are to be replaced by V'. While these solutions have been presented in terms of a quasiparticle theory, they are very closely equivalent to formulae which have been obtained from a somewhat different viewpoint by other authors 8-1t), viz. an expansion in an infinite series of eigenfunctions of I(E+ ie) or J(E). In this method, T for instance is expressed as an infinite sum over s of terms like the second term in eq. (2.29a), so that what we have called T' becomes the sum over all of these terms except the single term with s = 0. In preparation for the resonance development, we give below a collection of formulae concerning the eigenfunctions and eigenvalues of I(E+ie), J(E). The refer-

448

R. HUBY

ences for or derivations of these formulae appear in appendix 1.

(r ~ oo)u(E, r) ,., - V ~- (z(E)IVI¢°(E)) exp i(kr-½1n), [k =(e) (r -~ ~)v(E, ,') ~ -

//n/~

(z(E) I VIqSo(E))

fl(E)

cos (kr-½In),

Im ~(E) = - _~I(z(E)[ VIff°(E))[2,

(2.30a)

(2.30b) (2.31)

vl ,o(Z)> d (e) dE

J lim <~o(E)lV(E+ie-tlo)-2Vl¢o(E)>, (2.32) (~o(E)IVI~//o(E)) ~-.o

in which lim (~0(E)l V(E + ie- Ho) -2 VI~,o(E)) = ~2(E)((lo(E)l~to(E))cona.,

(2.33)

t:-'*0

where the suffix cond. on the scalar product means that this integral over r is only conditionally convergent, because of the oscillating behaviour of u(E, r) shown in eq. (2.30a), so that convergence must be enforced by, say, putting in a factor e x p ( - T r 2 ) , with 7 eventually vanishing.

dfl(E) __- _ 1 lim Re (qSo(E)lV(Z+ ie-Ho) -2VIq~o(E)), dE (~bo(E)l Vl~bo(E)) ,-,o

ct(E) - fl( E) - - i~r(z( e) l VI ~,o(E) ) (z( E)I VI (%( E) ) ,

(2.34) (2.35)

(¢o(e)l Vl¢o(e)) )

1 '

E-E'

2n2

+ I (e)l

[(z(E)IVI~,(E))I26(E-E').

(2.36)

3. The resonance approximation We consider the exact formulae for the resolvents F, G eqs. (2.28a and b) and for the T- and K-operators, eqs. (2.29a and b). These suggest that a resonance will occur at an energy which makes the denominator [1-~t(E)] or [ 1 - f l ( E ) ] become very small or zero. This has been observed by other authors [refs. 8-11)], but we shall pursue the consequences systematically. If one accepts the so-called "natural" dcfinition of a resonance energy as that complex energy at which TI(E) or St(E) has a pole, then this can be realized in the present formulation by continuing eq. (2.29a) to complex energy and asking for the value of E on the non-physical sheet which makes .z(E) equal to unity. However, the development proposed here is a rather dif-

449

SCA'I-'I'ERING A M P L I T U D E

ferent one which describes the resonance phenomenon without ever departing from real energies. Let us first work in the T-scheme employing eqs. (2.28a) and (2.29a). We make our definition of resonance energy as the energy Eo which satisfies 1 - R e ~(E0) = 0.

(3.1)

The essential physical condition for a sharp resonance to occur here is that Im ~(E0) << 1.

(3.2)

This condition can be expressed alternatively, using eq. (2.31), in terms of matrix elements containing ~o- A study of the trajectories of %(E) as function of positive E [ref. 4)] indicates that the conditions (3.1) and (3.2) can be satisfied at low energy for waves of/-value other than zero. This can be understood from the r.h.s, of eq. (2.31), because at low energy 7.(E, r) becomes small inside the range of the potential V provided that I is non-zero. The formula (2.30a) for the asymptotic behaviour of the quasiparticle function together with eq. (2.3l), shows that when the resonance conditions (3.1) and (3.2) are fulfilled u(L, r) has only a small amplitude outside the well. That is, the quasiparticle state is "nearly bound", as one would expect. If further one wishes to approximate F ' and T' in (2.28a), (2.29b) by their quasiBorn expansions, the Weinberg theory states that this is permissible provided that all the eigenvalues %(E) other than c~(E) have magnitude smaller than unity, and this will be satisfied at resonance, according to eqs. (3.1) and (3.2), provided e ( E ) is the eigenvalue of greatest magnitude. We now make a linear expansion of [1 - co(E)] in energy about the resonance point, writing

u(E, r),

1-~(E) ~

-ilm~(E°)-(E-E°)(dE)

E= Eo"

(3.3)

This will be developed with the help of eqs. (2.31) and (2.32). However, all the subsequent working is much simplified if one assumes that within the range of the potential the quasiparticle function is approximately real, so that for instance one can replace (~ol Vl~ko) by (~ol V[~0). This is intuitively justified if the wave function is thought of as "nearly bound", and the approximation is further validated in appendix 2 by a perturbation calculation. This approximation is, however, a source of error which is difficult to quantify, and is one reason for preferring the alternative standingwave theory. At this point it is conveneint to adopt a standard normalization of the quasiparticle wave function ~o(E, r) since this is not self-evident for a function oscillatory at large distances. It turns out that the formulae obtained are most perspicuous if one imposes the normalization

u(E, r)

lim <')o(E)l V(E+

ie-Ho)-ZV[(Vo(E)> =

1.

(3.4)

t:--~ 0

Under the resonance conditions (3.1), (3.2) one sees from eq. (2.33) that this is approximately equivalent to (~)o(E)l~,o(E))oo.~. = 1. (3.5)

450

R. HUBY

We shall regard eq. (3.4) as the natural normalization of the quasiparticle function. Tills can be understood because, if the wave function is large inside the potential with a small oscillatory tail, then in the normalization integral (3.5) the contribution from the tail is small, and the part of the wave function inside the potential is approximately normalized as for a bound state. A further point is that the normalization (3.5) makes t~(E, r) within the well very close in both shape and magnitude to the Gamow eigenfunction of H at the pole of S~(E), if the latter function is normalized in the way proposed by Berggren ~3) and Romo ~2), who use for this essentially the same normalization integral as (3.5). After the steps indicated, eq. (3.3) can be written

1-:c(E)

(3.6)

E-G+kiro

(¢o(e0)l VlOo(eo)) ' with Fo = 2N (Z(Eo)I Vl6o(Eo))[ 2.

(3.7)

The width Fo is indeed small if condition (3.2) is satisfied. Then with the same approximation the T m a t r i x element Tt(E ) becomes, from eqs. (2.10), (2.29a)

~(E) ~ r ; ( E ) + •

ro

.

27t(E-Eo+½iFo)

(3.8)

This is a reasonably acceptable representation of the scattering amplitude in the vicinity of a resonance, the first term being a slowly varying background, and the second the actual resonance contribution. However, if one constructs the scattering function S,(E) of eq. (2.11) one obtains

St(E) ~

E - E o - ½iFo E-- E o + ½iFo

2niT/(E).

(3.9)

This fails of unitarity, which is not surprising in view of the liberties taken with phase in the derivation. It is very difficult to attain unitarity in the T-scheme, and we therefore turn to the alternative/Gscheme. Here the "formal" resonance energy is defined as the energy E~ at which the denominator of K in eq. (2.29b) vanishes: 1 - / / ( E t ) = 0.

(3.10)

This is the energy at which the Hamiltonian H has an eigenfunction which coincides with the quasiparticle function qSo(E, r) and has a phase-shift of ~,-r. A linear expansion of [ 1 - / / ( E ) ] is now made about the resonance,

(3.11) It is again useful to specify a natural normalization of the quasiparticle wave function,

451

SCATTERING AMPLITUDE

now q~o(E, r), and for this we propose lim Re (~bo(E)l V(E + i e - Ho)- z VlqSo(E)) = 1.

(3.12)

e~0

This is similar formally to the normalization in the previous scheme, eq. (3.4), and it will indeed be similar in substance, if one supposes that qSo(E, r) and i/%(E, r) are very nearly the same inside the potential, both being "nearly bound" states. This conjecture is proved correct in appendix 2, provided the resonance is sharp. The normalization (3.12) again means that the part of q~o inside the potential, minus its small tail, is approximately normalized as a bound-state wave function. Now eq. (3.11 ) becomes, on using eq. (2.34)

(3.13)

E-E1

1-fl(E) ~ (- o(E,)lVlgpo(E,) ? , and the K-matrix element Kt(E) is, from eqs. (2.17) and (2.29b) El

Kt(E) ~ K;(E)+ 21,):~'E--E,''

(3.14)

F, = 27r(z(E,)I Vlc~o(E,)) 2.

(3.15)

with The matrix elements Tt(E) and St(E) are found by substituting eq. (3.14) in eqs. (2.18) and (2.11):

St(E) ~ 1 - i ~ K ; ( E ) - ½ i G ( E - E I ) -1 . 1 + i~Kf(E)+-}WI(E-E,)-'

(3.16)

This can be manipulated by defining a background phase-shift 6'(E) as

6'(E) = arg [1 -ircK;(E)],

(3.17)

leading to St(E) ~ e -~'~'~

1- k-G-A

+½it

'

(3.18)

where the effective width F is F =

F, 1 + ~'-[d;(e~)3 2'

(3.19)

and the level shift A is A = -~.rK;(e3.

(3.20)

The background matrix elements T[(E) and K/(E) in all these equations will both be replaced in the first-order quasi-Born approximation by (z(E)I V'Iz(E)). The final formula (3.18) is a very satisfactory one, displaying a resonance imposed on a background phase-shift 6', and being unitary. It is of the form expected for a

452

R. r~UBY

resonance on general grounds (see ref. ~), eq. 15), and corresponds in many details to the one-level formula with background of the R-matrix theory [ref. 2), p. 323]. It is an improvement on the previous formula (3.9) obtained in the T-scheme. However, the content of the two formulae is not very different if one assumes that 6', T[ and K[ are all small and the resonance is narrow. The resonance energies Eo and E1 as defined by eqs. (3.1) and (3.10) are quite close, since eq. (2.35) shows that the difference between ~(E) and fl(E) is almost pure imaginary. Moreover, the widths Fo and Ft defined by eqs. (3.7) and (3.15) are nearly equal, assuming that within the potential ~o and q5o are almost the same. Let us now evaluate the scattering wave function ~b(E, r) cq.(2.8), in the T-scheme resonance approximation. On using eqs. (2.28a), (3.6) and (3.7) and making the same approximations as earlier one obtains

t)(E,r) ~ [I+F'(E+ie)]z(E,r)+ V Fo-- ~o(E,r) , (3.21) r 2 ~ E-- E o + ½iFo provided the sign of ~o(E, r) has been chosen so as to make Re positive. The tirst term here represents the incident wave distorted by the effect of the residual potential V'. The second is the resonance component, which is a multiple of the quasiparticle function ~b'o(E, r), and satislies all the requirements set out in sect. 1. It is a nearly-bound wave function with a small outgoing tail, and its magnitude inside the potential becomes large at resonance if the width is small. A check on the consistency of the approximations is that, if one calculates the long-distance behaviour of O from eq. (3.21) with the aid ofeq. (2.30a), one can infer a value of T~(E) by comparison with eq. (2.12), and this does indeed agree with the formula (3.8). An alternative expression for the wave function comes from the K-scheme. The scattering wave function ~(E, r) as given by eqs. (2.19) and (2.15) is first manipulated into a more convenient form by using the exact quasiparticle expression (2.28b) for G and (2.29b) for K: 0(e, r) = [1 +F'(E+

ie)]'/.(E, r)

+-

[1. --fl(e)][l + x

e+ieZ

i,~K,(e)]<4o(e)[ Vl~o(E)>

- Vg~o(e,r)-inF'(e+i~)Z(E,r)

H0

,

(3.22)

where now F'(E+ie) is the reso/vent defined by eqs. (2.6) and (2.7) when Vis replaced by the residual potential V' of eq. (2.27b). One may approximate eq. (3.22) by using the resonance expansions (3.13) and (3.14), with the result eia'(~;)

x

[ E + i~-1 Ho V4o(E,r)-ix/~nF,F'(E+ie)z(E,r)],

(3.23)

453

SCATTERING AMPLITUDE

provided the sign of ~o(E, r) is chosen so as to make
0(E, r) ~ x,iE__E__~)i_+_~p t

\/1 +n2EK;(E)] 2

+

V?

I

2~ 4,o(E, ,') , (3.24)

where b(E) is the complete phase-shift, - a r c t g Kt(E), corresponding to eq. (3.14). This again clearly exhibits a background and a resonance term. both with perspicuous behaviour. One criterion of the goodness of an approximate scattering wave function is its normalization, which ought to be the same as that of the free waves, eq. (2.5), i.e. (,k(E)IO(E')> = ¢J(E- E').

(3.25)

If, for instance, one made an error in the value of the resonance part of 6 inside the well by adding an incorrect term which vanished rapidly at infinity, this would not alter the scattering amplitude, but it would upset eq. (3.25). Ihc normalization integral @'(E)[O(E')) can be calculated for the approximate wave function eq. (3.21), with the aid of eq. (2.36). On making the same approximations in the T-scheme as previously, one does find that eq. (3.25) is satisfied.

Appendix 1 This gives derivations of the exact formulae at the end of sect. 2, starting with eq. (2.30a). From the eigenstate equation for the function ~o(E, r) of eq. (2.23), viz.

OdE, r) -

1

(e)Ee +

Ho]

Vq, o(e, r),

(1.1)

on using tile expression for the Green function, O"I(E+ i~-Ho)-~l,') = -

X_ exp i k l r - , ' l , 4x [r-r'l

(1.2)

one can read off the asymptotic form of ~o, obtaining the radial dependence shown in eq. (2.30a). The derivation of eq. (2.30b) is similar. Eq. (2.31) was given by Weinberg *), with the factor rc however incorrectly omitted: the proof is indicated there. Eq. (2.32) was given by Meetz 8), eq. (5.3). It is readily proved by working first at complex E in the plane cut along the positive axis, and studying the equation for the

454

R. HL'~Y

eigenfunction ~ o: (,Po(E)I V(E - Ho)-' V[0o(E)> = "~(E)<¢o(E)] VI¢o(E)>.

(A.3)

On differentiating with respect to E, and noting that as well as ¢o being an eigenfunction of l(E+ic), the adjoint ~o satisfies (Co(E)[ V ( E - Ho) -' = cz(E)(¢o(e)l,

(a.4)

_(~)o(E)IV(E_Ho)_2VItPo(E)> = d=fE) (ffo(E)lVl¢o(E)>. dE

(A.5)

one finds

This becomes eq. (2.32), when E approaches the positive axis from above. Eq. (2.33) is a consequence of eq. (A.4). To prove eq. (2.34), one writes at positive energy the counterpart of eq. (A.3): lira ½(q~o(E)[ V[(E + ie - Ho)-I + (E - is - Ho)- ~] VI~o(E)) = fl(E)(cko(E)[ VI~o(E)). ~o (A.6) On differentiating this with respect to E, and using the eigenfunction property of 49o(E, r), one obtains lira - ½(dPo(E)IV[(E + ie- Ho)- 2 + (E-- i s - Ho)- 2] VI q~o(E)>

= d~(~) <~o(E)IVlOo(E)>, (A.7) dE

which is equivalent to eq. (2.34). Eq. (2.35) is obtained by cross-multiplying the eigenstate equations for Oo(E, r) and 49o(E, r). A formal proof of eq. (2.36) will be given which is cavalier in treating singular expressions, but which can be more rigorously verified. From the eigenstate property of Oo(E, r) and ¢o(E', r) one has

<0o(E')l v(E'- i~,- Ho)- '(E + is- Ho)-I ViODE)) = ~*(E')~(Z)<~0o(W')l~0o(E)>. (A.8) On writing 1 (E' - is - H o ) ( E + i~,- Ho)

E-E'+2iz

E--~+tto

E+ie-HoA'

eq. (A.8) becomes

Therefore (0o(E')IOo(E)> =

~)

<~'o(E')lrlOo(E)> P E - E "

izr,5(E-E')

.

SCATTERINGAMPLITUDE

455

The principal part term here agrees with the first term on the r.h.s, of eq. (2.36), and the delta-function term becomes the second term in eq. (2.36) on using eq. (2.31).

Appendix 2 PERTURBATION

THEORY

A perturbation theory will be given ill order to approximate the relation between the eigenfunctions $,(E, r) of I(E + i~) and ~bs(E, r) of J(E). A similarity transformation is first performed on all quantities by means of the Hermitian operator ( - V) ¢, as in sect. 3 of ref. s), so that for instance

~(E, r) = ( - V)~k(E, r). As in sect. 2, the normalization of eigenvectors is left open. Then anew as an eigenvalue of the transformed kernel

i( E + ie) = ( - V)¢I(E + ie)(- V)-~. The transformed kernel

$,(E, ,.). New

(A.9)

a(E) can be regarded (A.I0)

J(E) is Hermitian, having a complete set of eigenfunctions

i(E+i,)is related to J ( E ) by i ( e + i,) =

J(E)+:;:,

(A.11)

where

:; = in(- v ) ~ a ( e - n o ) ( - v)~.

(A.12)

We now treat 2 as a perturbation and we may use ordinary Rayleigh-Schr6dinger perturbation theory to obtain the eigenstates and eigenvalues of i from those of J, even though ~? is not Hermitian. In first order perturbation for the eigenvalues one obtains

<$~(e)f~l$~(E)> ~,(~) ~ I3,(E)+ <$,,(E)l$,(e)> fl,(E)-in2

(A.13)

( Cko(E)l Vlcko(E) ) This could alternatively have been inferred from eq. (2.35). If one takes the smallness of the last term as a criterion of the validity of perturbation theory, this turns out to be essentially the same as the condition of sharpness of the resonance level, eq. (3.2). The first-order approximation to ~o(E, r) is ~0(E, r) = 60(E, r ) + • s:#0

C,~p,(E, r),

(A.14)

456

R. HUBY

with =

<,5.,(e)wSt$o(E);,

_ - in(~.~(E)[ VIx(E))(z(E)IVI~o(E))

(A. 15)

V Ig)s(E)) A more reliable test for the smallness of the perturbation is that the n o r m of (~ o - ~ o) be small, i.e. 2 IC~IZ(~(E)Iq~(E)) << <~o(E)lq~o(E)). (A.16) s¢O By using completeness one readily finds that this is satisfied provided

ne << 1,

(A.17)

[//o(E)--~(E)]Z<4)o(E)lVI4,0(E)> taking flo and fl~ to be the largest and next-largest eigenvalues. However, this condition comes back again to be much the same as that of the sharpness of the resonance level, eq. (3.2), on utilising eq. (A.13) for l m x~(E). The transform of the adjoint eigenfunction ~o(E, r) is given by the expansion (A.14) with C~ replaced by its complex conjugate. In conclusion, we have proved that when the resonance is sharp all three functions Oo(E, r), ~o(E, r) and qSo(E, r) are nearly cqual inside the potential.

Note added in proof." Dr. M. A. Ball has pointed out that the normalisation (3.12) is only possible provided tbe signature of the quantity on the left-hand side, which is invariant under multiplication of q5o by any complex number, is positive. However, this will be the case provided the resonance is sharp, i.e. provided eq. (3.2) holds. References

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

R. Huby, Nuclear Structure, ed.. Hussain et al. (North-Holland, Amsterdam, 1967) 11 A. M. Lane and R. E. Thomas, Revs. Mod. Phys. 30 (1958) 257 J. Humbler and L. Rosenfeld, Nucl. Phys. 26 (1961) 529 S. Weinberg, Phys. Rev. 131 (1963) 440 R. Huby, Z. Phys. 218 (1969) 417 N. Mishima, Phys. Rev. 177 (1969) 2505 H. Feshbach, Ann. of Phys. 5 (1958) 357 K. Meetz, J. Math. Phys. 3 (1962) 690 A. Herzenberg and F. Mandl, Phys. Lett. 6 (1963) 288 H. Rollnik, Z. Phys. 145 (1956) 639 R. Phythian, Proc. Phys. Soc. 87 (1966) 171 T. Berggreen, Nucl. Phys. A109 (1968) 265 W. J. Romo, Nucl. Phys. All6 (1968) 617