«; P

REFERENCEICM/19

IBT3KNAL REPORT(Limitc: distri'bution)

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FINITE-RANK POTENTIALS IN THE TWO-BODY SCATTERING PROBLEM *

G. Pisent **

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

The aim of these lectures is a general and - as far

as possible - complete outline of the two-body scatter-

ing theory in the framework of finite-ran3< potentials.

Several topics particularly important for phenomeno-

logical applications - like s-wave separable potentials -

are derived as particular cases. The last section is

devoted to many-channel, finite-rank interaction, which

may be of interest if one of the two bodies - i .e. the

target - is a particle with internal structure.

MIRAMAEE - TRIESTE

March 197*+

* Lectures given at the Extended Seminar on Nuclear Phys ics , 17 September21 December 1973, ICTP, Tr ies te . .

Work supported in par t "by CNR and ~by INFN, Sezione d i Padova.

** Permanent address: I s t i t u t o d i Fis ica de l l 'Univers i ta di Padova, I t a l y .

4f

- 1 -

1. INTRODUCTION

The increasing interest of nuclear physics in sepa-

rable potentials is well known . The motivation

lies mainly in convenience (the Schrodinger equation

may be solved exactly), and more particularly in pos-

sible application to three-body or few-body problems .

Whilst most of/literature on this subject refers

to nucleon-nucleon interaction (as a preliminary stage

to analysis of nuclear structure), these lectures

aim to make particular reference to the nucleon-nucleus3)

phenomenology (as -a preliminary stage to possible

analyses of rearrangement collisions,in the framework of

an exact three-body theory ). This opens some not fully

conventional questions , as for example the excitation of

internal degrees, of freedom of the target, which is usual-

ly considered only -in the framework of local interactions

The optical potential is - by its nature - non-local .

Anyway, employment of loaal (Woods and Saxon) potentials,

gives quite good results. The non-locality effects are

id8)

7 )weak , and good resul ts are obtained by local-equivalent

potentialsLet us consider the following non-local but spin- and

(*)energy-independent potent ial , say, m coordinate representation

(1.1) <rVr'>=V(r,r').

Locality means

(1.2) <rVr1>=6(r-rl

where l/(r) is any short-range (R) well. In order to des-

cribe small non-local effects, we may write

(*) See the Appendix and the "Notations" at the end.

(1.3) <rVr'>=(/(s)W(t); s= (r+r • )/2 ; t = r"-r • ,

and choose for W, e.g. a gaussian form, namely

(l.t) W(t) = (/73)-3exp[~-(t/3)2~] .

This interaction is weakly non-local if 3<<R, and tends

to locality as 3 approaches zero.

Let us consider now the separable interaction

(1.5) <rVr •>=aiv(r)v(rl ) ; io = ±l ,

and compare (1.3) with (1.5). It is immediately recog-

nized that (1.3) mayvweakly non-local (it is 1//0 only for

t<3), but not at all separable (in r and r') s whilst

(1.5) is separable but highly non-local.

Furthermore,it is easy to show that the non-locality of

(1.5) increases with the distance. Now it is known that

the nucleon-nucleon interaction should be non-local at

short distancess while the nucleon-nucleus phenomenology

suggests a weak non-locality of the type (1.3). ;

Therefore the argument that employment of separable

potentials is suggested by physical requirements of non- i

locality is somewhat misleading. The main argument in ' >

favour of separability is convenience, and let us see ;

•briefly what this means: the separable interaction (1.5) j

may be written formally as a dyad f

(1.6) V

(*) The signum function u) defines the attractive (-) or repulsive(+) character of the interaction. Note that (here and in the follow-ing) we do not make explicit the strength.

- 3 -

Then the Schrodinger equation reads (see eq. (A.16))

(1.7) (E-h)<r|k+>=to<r [x>< x

This is a non-homogeneous differential equation. The differ-

ential operator in the homogeneous part contains only the

kinetic Hamiltonian (h), and the solution is straightfor-

ward. However the interaction (1.5) is too simple an object

in view of most phenomenological applications (in parti-

cular with an attractive potential (u--l) of that type,

only one bound state is possible J, Then the problem

arises of a generalization.

Let us consider first the expression

This is not substantially different from (1.6), since

the form factors <5F[x> have not been specified. But if we

write ;i

I I I(1.9) v»w I xi>*ii<x,=^ [ x i

> <x iiv»w I x i>* i i<x,^ [ x i

> <x ii i1,3=1 i=l j

this is nolonger separable (and therefore it is richer as internal

structure: more than 1 bound state is now possible), while

retaining the important property that the Schrodinger

equation is analytically solvable. Note that (1.9) is si-

milar to (1.8), except for the Kronecker symbol (i.e. the

information i=j, crossing from ket to bra) which destroysshall

separability. This is why we / reserve the word "separable"

to (l.S), calling the interaction (1.9) "finite range"j

although in current literature "separable" (or factorable)

is used in a more generalized sense and includes (1.9).

finite-rank potential (1.9) (which obviously

includes (1.6) and (1.8) as particular cases), will be

the pyotdtgoHtBt of the forthcoming sections.

As a final remark on the importance of this ob-

ject, it is worthwhile to remember that each "honest"

short range potential Chonest"means that the Schmidt

norm exists) may be approximated by a finite rank po-

tential i 0 ).

2. • SINGLE-CHANNEL, FINITE-RANK INTERACTION

2a. Formal theory

This section deals' with the scattering (between

two spinless and structureless particles), described

by the interaction (1.9).

Let us define the column-bra vector <x, and the

row-ket vector %> as follows;

(2.1) <x=<X2* * *

<XT

>c>= »I>I

in such a way that

(2.2)

is a (one-dimensional) operator, while

(2.3)

<X 1 I X 1 > < X 1 I X2>- • • <X 1 ! Xj>

< X 2 | X l > < X2 |X2> < X 2 | X 1 >

< X T | X 1 > < X T | X 2 > < X T i x T >i i A r

- 5 -

is a matrix of numbers.

Let us then evaluate the Moller (or wave- or dis

tortion-) operator 0, which transforms the plane wave

k> into the stationary scattering state k+> ,

(2.i+) k+>=fik>,

and satisfies the Lippmann-Schwinger equation

(2.5) fl=l+gVfl,

where

(2.6) g = - ^ — =U-r~-

E -h k -h

(*)is the free propagator .

Substitute eq. (2.2) into (2.5)

(2.7) fl

Then, left multiplication by <^ and rearrangement ' lead

to the following equation between column-bras;

(2.8)

where

(2.9a)

(2.9b)

(*). The paragraph devoted to "Notations" is particularly necessary for anunderstanding °f this section. Observe that the precise definition ofthe symbols k and E allows (wherever no confusion is possible) theomission of the energy dependence of the operators.

- 6 -

By s o l u t i o n of ( 2 . 8 ) w i t h r e s p e c t t o < x n , and substi-

tut ion of <%tl i n t o e q . ( 2 . 7 ) , one f ina l ly obtains

( 2 . 1 0 )

From the Moller operator, one derives the (formal) wave

function (see eq. (2.4)]

(2.11)

and the T matrix

\ k>,

(2.12)

or, more explicitly

=u> T x.>|M-'1 . .<x.,ij x L J1^ J

(2.13)

where use has been made of the identity

(2.14)

-theThe T-matrix for'separable potential (1.6) is im

mediately derived as a particular case (1=1)

( 2 . 1 5 ) - l-co<XgX>"

t h eThe T matrix for/finite-rank potential with composite form

factors (1.8) is a particular case of (2.15), namely

(2.16)

Ix.xx,ij.

- 7 -

theSome efforts have been made in/direction of (1.8),

which has more structure than (1.6) while remaining

simpler than (1.9) (no matrix inversion is necessary in

(2.16)) . In this way something may be gained, i.e.

in the spectrum of resonances, but the number of possible

bound states is obviously always limited to 1.

2b> The representation

What we have done up to now is completely formal. Now we

must represent V and T, and pass to explicit formulae.

This raises interesting questions about the nature of our

potentials and the possibilities of application to the

nuclear phenomenology.

To carry out such an analysis, let us start from

the (most natural) coordinate representation of the

(simplest) separable interaction (1.6). By an expansion of

the type (A. 9.) of the coordinate vector states, we may

write

<rVr'>=-%- I <r |tm><r£mjx> *

(2.17)

<X [r ' I'm'>< I'm' |£ '>.

This interaction is not rotationally invariant, because of

the angular dependence of the spherical harmonics. The po-

tential x><;X) being a scalar, cannot mix together different

angular momentum states. It must therefore "be written as

^ <f \im><rim\ x>< x Ir'zmX jim|f

(2.18) i m

Jim

(*) A matrix inversion is not dramatic in the two-body case, but may be-come cumbersome in the three-body problem.

- 8 -

with

C2.19)

The potential (2.18) is rotationally invariant

(since J>™(r) Y^Cr1) ̂ ^-V .(cosrr •) is a scalar product

between spherical tensors), &«t -tt I'S nolonger- separable.

In fact, for the sake of rotational invariance,V(r ,rM

musf depend on r,r' and rr' only. But separation ex-

cludes rr', and imposes the form (1.6) or (by compa-

rison between (1.5) and (2.18)) the form factor

(2.20) v.(r):/HV«. v(r). -

A full separable potential acts only'in' S-wave*

The important point is that, if V(r,rr) is not de

fined "in bulk", in eq. (2.18) the coefficients v are

completely uncorrelatnd. Observe that, for a local po-

tential, an expression similar to (2.18) may be written

by multipolar expansions of the 6 function, namely

(2.21)ton l r2

In other words the locality is sufficient to strictly

correlate the multipolar expansion coefficients, while

in the non-local case they may be uncorrelated. The un-

correlation means that v may be defined freely for

each Jt.' and is typically two-fold in the sense that

the models (which/' make1 ' use ' of these potentials)-

are very flexible for phenomenological purposes, but for

the same reason an explosion of free parameters can make

the model meaningless.

- 9 -

But there is another important consequence of un

correlation. Consider the transformation law between

(coordinate and momentum) form factors (see eqs. (A.12)

and (A.13) and Ref. 12 for definition of F )

(2.22) u £ ( p ) i ! !J

(2.23) X j u |

whose projection, e.g. on the momentum state,is

(2.24) < p | x A m > ^ U | 1 J

By means of ,the eq. (2.23), the potential (2.18) may be

written as follows;

Now the interest of the momentum representation is well t';•

known, two important motivations for this being the fol- *

lowing; }y

- the free propagator is diagonal in p representation; >,

- the p representation of T is (proportional to) the

scattering amplitude, leading directly to the final goal, of

the theory, i.e. the cross-section.

Then, since uncorrelation is maintained in momentum

space, one may define u^(p) directly, avoiding the compli-

cations of the transformation (2.22). However in this

case the centrifugal barrier (clearly present in eq. (2.22)

through F ) disappears, and the correct threshold be-

haviour of the partial wave scattering amplitudes must be

checked a posteriori.

Let us nov consider the interaction (2.18). Define

the state vector

- 10 -

If we put Jtmii and truncate the expansion (2.18)

at a certain maximum angular momentum, we go back for-

mally to (1.9), and this is a further check that multi-

polar expansion, together with rotational invariance, des-

troys separability. Nevertheless the situation in (2.25)

is simpler than in (1.9), because of orthogonality of

the set x£m>.

In fact

(2'26) <^J^, m,>=6 J U, j m m

It follows that x/g^ is diagonal,

( 2 . 2 7 a ) x i g )n . , = < 5 n f l , , X (

og )

Jim, I 'm ' U ' j B i m 1 A

( 2 . 2 7 b )k2+_p2

and M i s diagonal as well

This leads to great simplification of eq. (2.13), namely

C2-29) T s \ K n >\KnV t mJim

Now go to momentum representation and evaluate T on the

energy shell (off-shell extension should be straightfor-

ward )

(2 .30a) <kT(k)k'>=w I Y"J(k)T (k)YI!l*(k ' ) ,

(2.30b) TA(k)=uJ(k)/M£(k).

- 11 -

Then split M. (by standard procedures based on the Cauchy

theorem) dntQ real and imaginary parts(P means principal

value)

(2.31a)

(2.31b)k2-p2

This separation is important for deriving

ing matrix. In fact, start from

the scatter-

(2.32) = l-2iri6(E -E.)T,i -I*

sandwich eq. (2.32) between final and initial momentum

states (on the energy shell) and expand by means of eq,

(A.9)

(2.33) -k ' )S „ (k)<j£m|l

I |k2 Jim

Now compare with ( 2 . 30a)and( 2 . 32 ) and find

(2.34) TB(k) .

Finally, by ( 2 > 30b)and( 2 . 31) we get

(2.35) S.(k)=M*(k)/M.(k)

Now it is easy to specialize to (S-wave) full separable

potentials (2.20) and generalize to finite-rank ones (1.9)

In the first case'we obtain

- 12 -

( 2 . 3 6 a ) M 4 <k)=6 M 0 ( E ) = 5 £ [No ( k ) + 2a ju i r 2 i ku 2 ( k f | ,M 0 ( E ) 5 £

( 2 . 3 6 b )k2 + - p 2

In the second case (multipolar expansion of the fi-

nite—rank interaction) one deals with the composite space

£m,i. Nevertheless, since the X, g and M are diagonal in

the Jim, but not in the i space, it is convenient to se-

parate and define for example the row-ket vector _x >

as in (2.1) (vectors and matrices refer to i space

only) as follows;

(2.37a) ££m

(2.37b) <Pl

The X, matrix nov reads as follows

(2.38a) <X£

(2.38b) XJl

This demonstrates that the free propagator is diagonal in

the space £m as expected and allows us to write

(2.39a) tl£ = A,-w4S\

(2.39b) 4 g )

where we have introduced the new matrix

(2.40a) U (PP<)=u (p)u /(p»),

- 13 -

(2.40b) ^(p,p)=U

Now we are able to write down the transition matrix, both \

formally and explicitly in momentum representation [with {

reference to eq, (2.3Oa)),v i

(2.41a)

(2.4lb)

In eq. (2.41b) use has been made of the well-known property

(2.42)

The matrix LL is characterized by a set of very no-

ticeable properties, namely:

(i) all rows are proportional to the row [u (p')...u (p'))

so that the determinant is zero (jU (pp'J|=OJ;

(ii) for the same reason the determinant of any/is zero, •*x

cofactor / and consequently lL(pp')=0.

(iii) LiJ!>(p)=LJ!£(p): in the particular case of p = p* (eq.

(2.40b)) the matrix is symmetric (trivial).

On the ground of these properties, eq. (2.4lb>

may be modified as follows.

Separate ^ into real and imaginary parts, as in

eq. (2.36)

(2.43a)

(2,43b)

- 14 -

Then evaluate the determinant form (2.43), making use of

the properties (i)and(ii). The result is

(2.44) |MA

Finally, since

(2.45)

the transition matrix may be put into the form

(2.46) T =

where ji appears completely separated into real and imagi-

nary parts. Furthermore, application of (2.34) gives

(2.47a)

It is worthwhile to comment briefly on eq.-(2.4 7a)

- k is obviously meant to be real, otherwise S (k)=

S0(k)=|M0(k)r/|M0(k)|

II \ fi

- R and U (and consequent ly | J£ | and Tr(ys £[ )) a r e a l l ?even in k, so t h a t (see eq. (2 .44) ) we may w r i t e jj

(*) The demonstration (easy but not t r iv ia l ) is based on the follow- Ving rule for derivation of determinants(a) . . _ . - ' *

where P means derivation with respect to any variable, and Vk is thematrix whose elements are PA... Now, put x=d(rr/̂ u>pk and derive eq.(2.43a) with respect to x 1;3

(b)'

Then derive ||L | by means of (a), and make use of (b).'

(c) ( d / d x ) | M £ | Q £ ^ J ^On the other hand,derivation of (2.44) gives

and comparison between (c) and (d) demonstrates (2.45)

15 -

(2.47b)

- A deep analysis on the function M, with particular refer-

ence to the theory of the Fredholm determinant and of

the Jost function (both suitably extended to non-local

interactions), is interesting but out of the bounds of

these lectures 13^.

2c. The coordinate representation

Once the formal theory is set up, the problem of

coordinate or momentum representation is unessential and

the transformation from one to the other trivial, at least

in principle. There is, however, an important point, i.e.

the choice of the space in whichto define the form factors.

With connection to this point, it has been observed already

in Seel that, although form factors are traditionally de-

fined in momentum space with obvious motivations, it is

important to examine the alternative procedure , in

virtue of the following considerations:

- The effects of the centrifugal barrier are so implicitly

(and certainly correctly) taken into account.

- One may introduce some correlation between different an-

gular momentum states, in order to reduce the number of

free parameters.

- The introduction of Coulomb forces may be easier on the

ground of the coordinate representation

The starting point is the coordinate representation

of the Green function,which in the notations of the Appen-

dix reads as follows

- 16 -

where F (0 ) is the regular (outgoing) solution of the

radial free Schrodinger equations as defined in R'ef.12

and where r< . (r ) indicates the larger (smaller)

of the numbers r and r1. By insertion of (2.48) into

(2.39b), we obtain

(2.49)

xF2Ckr')+FA(kr) v̂ (r») r 'dr • 0£(kr • T} .

By operating on the eq. (2.49) with the following identity?

a(r)rdr b(r')r'dr•+ b(r)rdr a(r')r'dr•=

(2.50)

= a(r)rdr• b(r)rdr,

o othe matrix M may be put into the form (2.43a), where N.

and ^ are now expressed through integration in ordinary

space; namely

N*,ijrdrv .(r)G0(kr)*13 k

(2.51a) 'Ir'dr'v, .(r')F,(kr')+[rdrv. .(f)G.(kr)-

•Ir'dr'v. .Cr')Fn(kr')

(2.51b) U rdrv. .(r)F.(kr)• rdrv. .(r)F.(kr)

-17 -

Observe that: ,

- In eq. (2.51a) use has been made of the relation 0 = •

=G.+iF , G being the irregular solution of the radial JX X* Xi r

free Schrodinger equation (see Ref. 12 . ) • \x

- In eq. (2.51b) the transformation lav between Icoordinate and momentum form factors (2.22) is \immediately recognized.

- It is obviously intended that the form factors v (and

the functions a and b of eq. (2.50)) obey the condi-

tions which ensure the existence of all the integrals

considered up to now. Without details on this point, we

observe only that integrals containing the irregular

solution G5 may give trouble at the origin. However it

is possible to demonstrate (see Ahmad and Pisent in Bef.

14.) that the regularity of F is sufficient to boundyi "

/a(r)drG (r)/b(r')dr'F (r'), provided a and b do notQ. * 0 *

diverge.

Now let us write down the wave function, whose co-

ordinate representation is certainly more important than

the momentum one. Take eq. (2.11) and project on <r

<r |k+>=<r |k>+ u) I <r|g(k) |r'>

(2.52)

dr'<r' I Y > M~ •<)

Then expand all state vectors following the Appendix and

the eq. (2.48)

(2.53) 0

+Ffl(kr)|oo(kr')v0 .(r')r'dr'

0&(kr)j^(kr')v£(.

( MI l )ij ( k ) *

-coF0(kr') v. ,(r') r' dr' .

0'

- 18 -

Let us give finally the expression of M and ifr for

the particular case (2.18).'

rdrv0(r)On(kr)b'dr'vn(rt)F0(kr

t) ,

(2.55)<K(k,r)=F0(kr)-- kMnTk)

+F.(kr) O.(krf)rldrlv,(rt)

On(kr) rldrlFft(kr')vn(r')+

CO

r' dr1 v£(r')

3. MULTI-CHANNEL, FINITE-RAM INTERACTION

3a. Formal theory

We consider now the scattering of a structureless

projectile by a composite target. The philosophy is to

assume that the spectrum of the target eigenstates is

known, so that all inelastic channels may be treated

exactly (within the limits of validity of the model which-

describes the internal structure of the target). This is ,

the so-called "generalized optical model", which has been

stressed in the case of local potentials, up to a very

sophisticated stage of phenomenological application

Our aim is to reformulate the whole problem in the

framework of finite-rank interaction, trying to evidence

analogies and differences with the local case, with par-

ticular reference to the "compound resonances", i.e. those

resonances which are generated by virtual excitation of

the target eigenstates.

Let us proceed in a very standard way, by writing

the total Hamiltonian H as follows:

- 19 -

( 3 . 1 ) . H=h+h +V,

as a sum of the kinetic Harailtonian h, plus target Hamil

tonian ht .plus projectile-target interaction V. We as-

sume that h has a complete, orthonormal set of eigen-

sta tes 01 ^

(3.2) h a)=e a)t a

(3.3) ( a l a ^ = 6 a a . J E a*>Co = l,.

Our conventions are that a=l means elastic channel, and

that zi-0. We indicate by E (k) the energy (momentum) of

the incoming particle

(3.4) h|k>=E|k> ,

while the channel energy (momentum)

(3.5) E = E-e ; Q2=yE ,

is defined in such a way that ot,q > ("belov the threshold

a) is the eigenstate of the unperturbed Hamiltonian (h+h =

sH-V), with eigenenergy E

(3.6) (h+lv )|aq >=(E +e )|aq >=E]<xq >.

At this stage we introduce the unconventional element, i.e

the projectile-target interaction dependent on the target

state. This can be realized by writing formally as in eq.

(1.9)

(*) The round ket a) refers to the internal (target) space, while thesharp ket is reserved either to the external (relative motion) space[k>], or to the tensor product of the two spaces [a,k>] .

- 20 -

(3.7)

with the fundamental difference that V is now defined in

space ak> and is ct-dependent

(3.8)

Now the formal treatment may be carried out in close

similarity with that of Sec. 2, only beirig careful that

- the unperturbed state is now |a=l,k> ,

- the unperturbed propagator is

(3.9) G(E)=E+-(h+ht) E

+-(H-V) .

So, for example, eq. (2.4) is substituted by

(3.10) k+>=fl|l,k>,

but Moller operator and transition matrix have the same form,

namely

(3.11) n 1

(3.12) M=1-O

(3.13) T=VO.

It must be pointed out however that, in spite of the

close similarity between the two formal treatments, the

next section, devoted to "representation", will evidence

the fundamental differences in the physical pictures.

- 21 -

3b. The representation

We consider here the case of spinless - although not

structureless - particles. The properties of the new space in

which to project, are immediately derived from eqs. (3.3)

and (A.2), namely

(3.14)

The mul t ipo la r expansion does not imply new problems (with

respec t to Sec. 2) s ince £m s t a t e s a r e factorized anyway.

The form fac to r ( a f t e r mul t ipo la r expansion) i s now

(3.15) <P<tm|x a i >=u £ > a j i (p) ,

while eqs. (2. 37) , . . . , (2.39) are substituted

lowing:

by the fo l -

(3.16)

.17a)

(3.17c)

(3.18a) MJl

(3.18b) 4E+-<

Ufl (p)p2dp

- 22 -

Now we are able to write, down the matrix element of

the (on-the-energy-shell) transition matrix, in the sub-

space of open channels

(3.19a)

( 3' 1 9 b )

Xr ill

Observe that M means, as before, matrix in the i (tensorial

rank) space, while T is'a matrix element in the space of

the target eigenstates. U o is a very particular object,—op

active in both spaces.

Let us concentrate our attention on the matrix M

(eqs.(3.18)). The sum over a, coming from completeness,

runs over all (open and closed) channels. But if a is open

(E>£ ) q2 is positive and the integral in eq. (3.18) is

complex, whilst for closed channels (q2<0) the integral is

real. The separation of ££ into real and imaginary parts is

now a little cumbersome, but physically meaningful,

closed a

(.3.20)

uI ,Q2+P

open ot-P iccVac/V

where Q indicates the imaginary part (if any) of q .

The scattering matrix relative to the open channels

is defined as follows;

- 23 -

From eqs. (3.19), (3 . 20)and( 3 . 21) it may be recognized

that:

- The imaginary part of ]£ resfers to open channels only

(note that at least/elastic channel is open in any case).

The consequence is that S is unitary in the subspace of

open channels (closed channels cannot be directly excited).

- Nevertheless, significant elements of T and S (relative

to open channels) contain (through the real part of tl)

all (closed and open) channels, and this makes virtual

excitation possible, as it will be better seen later.

Let us consider now in more detail the matrix element

S , which is sufficient to get elastic and (overall) in-X. , 1 1

elastic cross-section. It is convenient to write again eq.

(3.20) in a form similar to (2.43), namely (remember that

(3.22a) M(k)=N0 , (k) + i(ir/2) kU (k)

closed a

(3.22b)

open a

(p)p2dp

open and

The motivation for this is that iL ..(k) in eq. (3.22a)

satisfies the same properties as the matrix U (k) in eq.

(2.43). As a consequence we may arrive at the following

equations, formally equivalent to the set (2.44), (2.45),

(2.46);

- 24 -

(3.23) T.

(3.24) SA

(3.25) |tt

There is, however, a substantial difference: only below the

1st threshold, N̂ is real and S, unitary. Otherwise

it is |S i<l, and the unitarity must include the other)L y i 1

open channels. This is the reason why we have adopted for

S the form (2.47b) (of general validity) instead of (2.47a),

which now holds only in the elastic region.

3 c . The compound resonancesr

Let us recall briefly what is meant by "composite

resonance" in a coupled channel problem: switch off all

couplings and suppose that the (closed) channel a exhibits

a bound state at energy E = Eo~e (<0). Then, if all couplings

are weakly switched on and the projectile impinges

with energy E^EQ=E + e , it excites the target, looses e

and is quasi-bound with the remaining E to the excited tar-

get. The consequence is a narrow resonance in the elastic

channel.the

All this is .correct if /couplings are sufficient to

enhance channel a to 1, but not too high. In fact,if the cou-

plings are increased, the whole problem cannot be considered

in a perturbation philosophy, the resonance enlarges and

leads to a loss of parenthood.

Let us return to our model and consider the Schrodinger

equation

(3.26) (H-E)|k+>=0.

• ? • • • '

- 25 -

Expand k+> in target eigenstates

(3.27). k+>=£ a)(a|k+>.

Substitute (3.27) into (3.26) and project on the state (el

(3.28)+ "Ii I (0|xi><Xila)(a|k+>=O.

When projected on the momentum, this becomes the coupled

system which we have already implicitly resolved. In ac-

tual cases the expansion in a is usually truncated at some

a=A on the ground of physical considerations, so that the

order of the system (3.28) is finite. In comparison with

similar systems in the coupled-channel local problem ,

the set (3.28) is characterized by complete mixing among

diagonal (optical) and off-diagonal (coupling) potential

elements. In other words, the unperturbed situation (all

couplings switched off) is characterized here by complete

absence of background in all (closed and open) channels. ;

This amounts to saying that it is impdssible to ascertain(*) '

paternity here . {

The obvious way to obtain.a "weak coupling" model is to ]

add central local potentials. Otherwise, within the finite- j

rank' interactions, it is necessary to correlate in some way ,

the space i with the space a, as is done in Ref. 18 . |

ACKNOWLEDGMENTS

I should like to thank Professor L. Fonda for discussions about some critical questions dealt withhere. I should also like to thank Professor Afodus Salam, the International Atomic Energy Agency andUNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

The system (3.28), in the particular case 1=1, A=2, is discussed also inthe paper Ref, 17 » with particular reference to the "compound reso-nance" problem.

- 26 -

APPENDIX

In this i pendix.all state vectors introduced are properly defined.• • ; t h e

Although some of these are very familiar in/literature, a general

survey of the definitions, with particular reference to the normaliza-

tions , is convenient.

The eigenstates of the free Hamiltonian (plane wave) give the

connection between coordinate and momentum representations

(A.I) . <

Coordinate and momentum vectors are both orthonormal and complete

(A. 2) <p"|p'>=6(p-p');

(A, 3) <r \vl>= 6(r"-r'); I r>d"r^=is:

as it may be immediately verified by means of

(A.4) exp(ip',r)d'r=(2TT)36(1p).

We introduce then the eigenstates of the angular momentum operator,

i.e. the spherical harmonics, which define the connection between an-

gular coordinate (or momentum) and angular momentum states

(A. 5) <r| Am>=YI!1(r); <p j £m>=Y™(p). <

and iProperties of the type (A.2) /(A.3) hold in this case too, but dual «

spaces are not symmetric (r is continuum and £m discrete) \

(A. 6) <£m] Jl'm'^fi t ,;* Am

(A.7) <?jr'>=6(r-r'); |f>dr<r=i,.

- 27 -

The explicit (non formal) orthonormality and completeness read

* £m

19)

Less trivial (and less known in literature ) is the radial (coordi-

nate and momentum) state vector: let us expand, say,the momentum ket

by means of the completeness (A.6) (expansion from ket to bra and from

momentum to coordinate is obvious)

(A.9) p > = _Jim v

The meaning of the ket pJtm> i s found by expanding the plane wave, f i r s t

formally

(A.10) <r|p>=— I <r\ln><x>lm\vl'm'><l'ml\p>," Jim

Jl'm'

and then explicitly

(A.11) . (2Tr)"^exp(i^r) =J-— J i°P A

&m

By comparison we have

Now take the completeness (A.3) and expand ket and bra by means of (A.9)

It is easy to eliminate the angular part by means of (A.8), and one gets

the "radial" completeness

(A. 13)

On the other hand,the orthogonality

(A.14) ^Jlmlr'A'm'^fi^, mm,6(r-r()

- 28 -

may be deduced from the well-known normaJLization of the regular so-

lution F, namely

(A.15) JF^prJdpF^pr' )=^6(r-r').

The radial states are nothing but eigenstates of the radial free

Schrodinger equation (after angular momentum separation). It is easy

to check the coherence of all definitions, in the sense that decompo-

sition of the first equation in (A.3) by means of (A.9), and employ-

ment of (A.14), leads to •

(A. 16) 6(r"-r')=6(r~rl) 6(r-r').

We consider,finallyjthe eigenstates of a complete Hamiltonian,

and more precisely the ket p+>, relative to the stationary scattering

state (if any), defined by the Lippmann-Schwinger equation

(A. 17) -p+>=i>i J- Vp"+>.E -kP

Although the state obviously depends on the (central) interaction V,

some formal development may be carried out in strict analogy with

that of p>. Namely, in coherence with (A.9), let us write

(A.18) p+>= I p+£mX£.m|p>.Jim • P

The ket is defined as usual by projection on coordinates,

(A. 19) <r|p+>=— I <5;|£m><rJlmlp+S.'mt><)llni' |p>.^r £m

Z'm'

In particular it is convenient to write, just as in eq. (A.12);

(A.20)

- 29 -

in order to deal with a radial solution (ij> ) which approaches F. as

the interaction is switched off. From eq. (A.17) it is immediately

shown thatr

_ , _ f r )(A.21) <p+|p'+>c<pjpt>s6(p-pt) p+>dp<p+= p>dp<p=l! , .

and consequently ^

(A.22)

NOTATIONS

r,k means vector in ordinary (coordinate or momentum) space

P,k are corresponding unit vectors

dr=r2drsin-&d§d<p; d?Esin%d-9d<p are differentials referringto total (3 dimen-sional) or angular space

2( is a vector in the space (spanned by i=l,2,...,1), defined by thetensorial interaction (1.9)

!£ is a (square) matrix

"M* is the transpose (M =M )~ * mn ^ mH is the adjoint matrix (M =cofactor of M with sign)*" . mn mn °TrH=5! ̂ is "the trace= •'n En||lj means determinant of ̂ (observe that the double horizontal line dis-

tinguishes the determinant |MJ from a vector modulus |r"|)

M"1 is the inverse of the matrix ̂ (note the difference between theelement of the inverse matrix (M"1)..=M../JMJ and the inverse ofthe matrix element MT^)

U=2m/Vi2; m=reduced mass

E and k=/ilE are kinetic energy and momentum of the system of particlesunder consideration in the centre-of-mass reference

E and p= AiE are used as "variable" energy and momentum

E and q -SuE are "channel" energy and momentum

/ 1 i all integral and summations are intended to be extended to the wholespace under consideration, unless otherwise stated.

-30-

KEFEKENCES

1) See, for example, K.M. Watson and J. Nuttall, Topics

in Several Particle Dynamics, (Holden-Day 1967);

J.S. Levinger, "Two- and three-body problems", Lec-

tures held in Orsay (1973), unpublished, and papers

quoted therein•

2) A.N. Hitra, in ' Advances in Nuclear Physics 3_ (1970);

J. Gillespie and J. Nuttall, Three-Particle Scatter-

ing in Quantum Mechanics,-(1968).

3) A.TJ. Mitra et al., Nucl. Phys. _3JD, 257 (1958);

P. Beregi and I. Lovas, Phys. Lett. 3_3, 1 5° (1970) )

J. Pigeon et al., Nuovo Cimento 1A, 285 (1971);

J. Pigeon et al., Phys. Rev. CM_, 704 (1971))

P. Beregi et al. , Ann. of Phys. j5_l, 57 (1970).

I) P. Beregi and I. Lovas, Zeit. f. Ph. 211, 110 (1971).

5) G. Pisent,in Theory of Nuclear Structure: Trieste

Lectures 1969 , (IAEA, Vienna 1970).

6) See, for example,H. Feshbach, Ann. Phys. 5_ 357 (1958).

7) F-.G. Perey and B. Buck, Nucl. Phys. Z2_, 353 (1962).

8) D. Wilmore and P.E. Hodgson, Nucl. Phys. 5_5_, 673 (1961).

9) F. Tabakin, Phys. Rev. _1, 177 (1969).

10) See,for example,C. Lovelace,in Strong Interactions

and High-Energy Physics, (London. 1<9 61) p. 137.

II) P. Beregi, Lett. Nuovo Cimento 2_» 233 (1971).

12) A.M. Lane and R.G. Thomas, Rev. Mod. Phys. _30_, 257

(1958).

13) M. Bertero-et al. , Nuovo Cimento M_6, 337 (1966); I

C.S. Warke and R.K. Bahaduri, Nucl. Phys. A162, 289 (1971)/

-31-

1^) W.H.S.J. Nichols, Amer. Jour, of Phys. ̂ 3, 474 (1965);

R.L. Cassola and R.D. Koshel, Jour, of Phys. 1A, 224

C1968)>

S.S. Ahmad and G. Pisent: "On the separable potential

in coordinate representation", Nuovo Cimento (in Press).

15) S. Ali et al., Phys. Rev. D6, 1178 (1972).

16) See, for example,, P.E. Hodgson: Nuclear Reactions and

Nuclear Structure, p.873,and papers quoted therein.

17) L. Fonda, in Fundamentals in Nuclear Theory. (IAEA» Vienna

1967) p.. 793.

18) G. Pisent and S.S. Ahmad, Nuovo Cimento 12_, 665 (1972).

19) C.W. Wong and D.M. Clement, Nucl. Phys. A18 3 (1972).

n