Computing matrix elements through inhomogeneous equations in multichannel problems

Abstract

Matrix elements are closely connected with solutions of inhomogeneous Schrödinger equations. Based on the recently developed renormalized method for the solutions of multichannel inhomogeneous Schrödinger equations, simple procedures are derived for the calculation of first-order and second-order matrix elements in multichannel problems. First-order matrix elements are associated with the asymptotic outgoing solution of an inhomogeneous equation and their values are given by the renormalized method directly. For a second-order matrix element, by writing it first as an overlap integral with the outgoing solution of an inhomogeneous equation, recursion relations for two auxiliary quantities, which completely determine the overlap integral, are then derived. These recursion relations should be propagated together with the renormalized solution. The value of the second-order matrix element is given at the end of the propagation. The algorithms are illustrated by numerical examples.