Algebraic derivation of the generalized Talmi expansion

_~icthods which serve to evaluate matrix elements of two-body interactions in a single-particle basis are applied in most branches of the physics. The most widely used single-particle basis is that of the harmonic oscillator cigenstates. ]"or interactions depending on the relative co-ordinate of the two particles, the Talmi expansion, which expresses the product of two harmonic oscillator wave functions in terms of relative and centre-of-mass dependent functions, is widely employed (1). The generalized Tallni expansion involves the product of two harmonic-oscillator wave functions with different well parameters (2), In the present communication the product ~,,(x~)~,,(~x2), where ~0,, are harmonicoscillator wave functions, and ~ %/im2w2)/(mlo~l), is expanded in terms of relative and generalized centre-of-mass co-ordinates dependent harmonic-oscillator wave functions, by a strictly algebraic t reatment . Use is made of the fact that 1) transition to relative and center-of-mass co-ordinates is obtained by a rotation in the xl-x 2 plane (3), and 2) the well parameter is affected by scaling of the corresponding co-ordinate. The product space {r n l , n 2 : 0 . . . . . ~} is isomorphic with that of a two-dimensional isotropie harmonic oscillator in the (x-y)-plane. l(otations in this plane 9 f t are generated by L z : ,(axa~--a~a,), where a, and a~ are harmonic-oscillator annihir , t t Jr lation operators. The three operators J l : (a~a~ma~a~)/2, J2=.Lz/2, J 3 : (a~a~--axa~)/2, which commute with the Hamil tonian of the two-dimensional isotropic harmonic oscillator, are the generators of SU, , and obey the usual angular-momentum commutat ion f relations. They commute with j2 = j~ + j~ HJ~ ~ (Ar/2)(N/2 -~ 1), where N = a,a , -~ ~-aCvav, qSn~(xl)qJm(x~): ]nx, n~} is an cigenstate of j2 and Ja, with eigenvalucs ](]"-k 1) and ~3 such that j = (n~ .--' n~)/2; ~'a = (n~--n~)/2. Therefore, In~, n~} _: 1], is). Define a transformation T=S~(~)R~(O/2), where S~(~) is represented in the (x-y)-space by the matrix