Solution of the Lippmann-Schwinger Equation with an Expansion of the Wave Function inside the Potential Range

Abstract

We propose a simple but rigorous formalism to solve the Lippmann·Schwinger equation, expand· ing the wave function inside the potential range. The solution gives us a separable t-matrix. The formalism can be applied to any finite range two-body interactions and is useful to solve few-body equations. The solution of the two-body Lippmann-Schwinger equation is important not only in the two-body problem alone but also in the few-body problems. It is well known that a t-matrix solution in a separable form is very useful in the few-body equations. l ) Here we propose a new formalism to solve the two-body Lippmann-Schwinger equation. The assumption is that the two-body interaction is of finite range, i.e., vanishes beyond the range R. We expand the wave function I¢) inside the potential range. Since the range is finite, it is easy to find a set of complete base functions for the expansion. The expansion is useful in the integral equation because it keeps the product Vl¢) unchanged. Then we get a set of linear equations each for the scatter­ ing problem and for the bound state problem. It will be shown that the solution of the scattering equation leads to a separable t-matrix. Thus the present formalism is very useful in the few-body equations. When it is applied to the two-body bound state problem, it easily gives the bound state wave function in the whole range, although the original expansion is limited inside the potential range. The resulting separable t-matrix in the present formalism can also be derived from a separable potential expansion method developed by Adhikari and his collabo­ rators. 2 ) Their method has been used in a recent nod scattering calculation by the present author3) adopting modified Malftiet-Tjon potential as the two-body interac-. tion. 4 ) It has been demonstrated that the phase shift and the absorption parameters of the nod scattering are obtained very accurately with low rank separable potentials. The result has been compared with those obtained from other methods. 4 ) It was found that the agreement between them are satisfactorily good. However, it is sometimes asked why a separable potential expansion works so well. The present formalism gives a simple but solid theoretical base for the nod scattering calculation done in Ref. 3). First we discuss the two-body scattering problem. Hereafter, we use the partial wave projected one-dimensional equation, for simplicity. But, it is not essential. We could also use a three-dimensional one. The scattering wave function j¢) satisfies the Lippmann-Schwinger equation,