Solution of the scatteringTmatrix equation in discrete complex momentum space

Abstract

The scattering solution to the Lippmann-Schwinger equation is expanded into a set of spherical Bessel functions of complex wave numbers, ${K}_{j}$, with $j=1,2,\ensuremath{\cdots},M$. The value of each ${K}_{j}$ is determined from the condition that the spherical Bessel function smoothly matches onto an asymptotically outgoing spherical Hankel (or Coulomb) function of the correct physical wave number at a matching point $R$. The spherical Bessel functions thus determined are Sturmian functions, and they form a complete set in the interval 0 to $R$. The coefficients of the expansion of the scattering function are determined by matrix inversion of a linear set of algebraic equations, which are equivalent to the solution of the $T$-matrix equation in complex momentum space. In view of the presence of a matching radius, no singularities are encountered for the Green's functions, and the inclusion of Coulomb potentials offers no computational difficulties. Three numerical examples are performed in order to illustrate the convergence of the elastic scattering matrix $S$ with $M$. One of these consists of a set of coupled equations which describe the breakup of a deuteron as it scatters from the nucleus on $^{58}\mathrm{Ni}$. A value of $M$ of 15 or less is found sufficient to reproduce the exact $S$ matrix element to an accuracy of four figures after the decimal point.