LETTERE AL NUOVO CIMENTO VOL. 9, N. 10 9 ~ r z o 1974

Derivation of the Mitra Three-Body Model from the Amado Theory.

V. K. SI~AI~MA

Department o] Physics, D. S. College - Aligarh

R. S. KAUSHAL

Department o/ Physics, Ram]as College - Delhi

( r icevuto il 13 Novcmbrc 1973)

1 . - In recent years some a t t e m p t s (1.5) h a v e been made to clar ify thc connect ion be tween different formula t ions of the th ree -body problem. These studies provide a convenient s ta r t ing poin t for discussing the general quest ion about the most sui table formal ism for more than th ree -body problems.

The th ree -body formula t ions which are commonly known are: i) Faddeev theory (6), ii) A m a d o model (7) and iii) Mit ra ' s formal ism (s). The first two approaches, in which the emphas i s is nlore on the react ion ma t r ix ra ther t h a n on the wave funct ion, are re la ted under some special assumptio'ns o'n t he two-par t ic le in teract ion ~ccording to Lovelace (9).

The Mitra th ree-body model has been re formula ted in the con tex t of the Faddeev theo ry in the separable-pe ten t ia l app rox ima t ion (5). In Mitra 's fo rmal i sm one s tar ts f rom the th ree -body Schr6digcr equa t ion and obtains effect ive two-par t ic le equa- t ions in the two-body subspace of t he so-called ~ spec ta tor funct ion )~. Here a reduc- t ion of t he Amado approach into Mitra ' s model for the on-energy-shel l case is shown.

2. - In the Amado model, the integral equa t ion for the n-d scat ter ing ampl i tudc in the c.m. system can be wr i t ten ~s (7) (by pu t t ing h = M = e -- 1)

(1) 1 3 1 ,

(1) P . A. KAZAKS a n d K . R . GREIDER: Phys. Rev. C, 1, 856 (1970). (~) T . A . 0sBom'r a n d K . L. KOWALSKI: Ann. o]Phys., 68, 361 (1971); K . L . KOWALSKI a n d S. C. PIE- PER; Phys. Rev. C, 5, 324 (1972). (3) V. VANZ~NI ~nd G. CATTAPAN: Lett. Nuovo Ciraento, 1, 1057 (1971). (4) Y. HAHN a n d K . M. WATSON: Phys. Rev. A, 5, 1718 (1972). (a) V. VANZANI: p r c p r i n t (1973). (9) L. D. F~I)DEI~V: Ma~hema~ieaZ Aspect o] Three-Body Problem in Qcea~um Scat~eri~.g Theory ( Je ru- sa lem, 1965). (7) R . A~,IADO: Phys. Rev., 132, 485 (1963); I. DUCK: Advances in Nuclear Physics, Vol. 1 (New Y o r k , 1968), p. 341. (~) A . N . 5IITRA: Nucl. Phys., 32, 529 (1962); Advances in Nuclear Physics, Vol. 3 (New York , 1969), p. 1. (') C. LOV:ELACE: Phys. Rev., 133, B 1225 (1964).

400

D E R I V A T I O N OF THE MITRA. T H R E E - B O D Y MODEL FROM THE A1VIA.DO T H E O R Y 401

where the Born amplitude is given by

(2) <pIB(E) Ik> = ~oO([p + �89 + �89

E + i e - - � 8 9 1 8 9 1 8 9 + k) ~ '

with FoO(q 2) ~/~(~2§ q~)T(q2). Hcrc the factor ~/2 arises from the identity of the two n 's and T(q 2) is the Fourier transform of the D-bound state wave function.

In eq. (1) the propagator is

( 3 ) 1 (3) p E - - - ~ q 2 = .

(E -? i s - - ~ q 2 -? ~2)

-2M" M = l) for (p]t(E)lk) on the Now substituting - - ( 2 z / t t ) a ( p ) (wherc tt = 3 , energy shell and using the deuteron ground state T (p) = Ng(p)/(p2~ - a ~-) in eq. (1), we get

3 (2z~) 3 4~ e g(p + �89 k)g(k + ~p) + (4) - - - a(p) = 4 N s ( p 2 - ~ k z + p ' k - E )

f g(p + �89 § �89 kS_ § d q ( ~ + q T ~ p _ q ~ E ) a ( q ) ( q 2 - - is) ~ ,

where iY is the normalization constant such that

(5) 1 f g2(q) N-2- - (2~)3 dq'(~2§ q2)~"

It can be shown that in the on-energy-shell ease H(p) defined by )/[ITRA (it) reduces to

(6) I (p2 _ k2)-~(x-1 _ h(p)),

H(p) = 3 (2~) 3

4 212

Substituting (6) in eq. (4) gives the following integral equation for a(p), exactly the same as given by MITRA:

(7) a(p)H(p) = 4z2K(p , k) § 2fdqK(p, q)a(q)(q 2 - - k 2 - - ie) -1 ,

where

(8)

and

K ( p , q) = K(q , p ) = D - l ( p , q )g(p § �89 ~- �89

(9) D-I (p , q) = (p2 + {12 ~_ p .q _ E) - I .

Thus we see tha t the integral equation in the Amado model is exactly reduced to that of Nitra 's one written in terms of the scattering amplitude a(p). Mitra's formalism

4 0 2 V . K . StIARM.A. and R. s. KA.USHAL

of three-body problem provides the concept of Wheeler 's resonating-group substruc- tures. Thus i t can be extended to problems of more than three bodies by using the resonating-group method with nonlocal separable potential . In more complex nucleus, however, one gets pract ical difficulties, since in resonating-group calculations one em- ploys a completely ant isymmetr ie wave function, which, from a computat ional point of view, is feasible only far systems with a relat ively small number of nucleons.

The resonating-group method with nonlocal separable potent ia l will be pre- sented elsewhere.

One of the authors (VKS) is grateful to Dr. J. H. NAQVI for his s t imulat ing in- terest in the subject of this work.