A “canonical functions” approach to the solution of two coupled differential equations of scattering theory

Abstract

The system of two coupled Schrödinger differential equations (CE) arising in the close-coupling description of electron-atom scattering is considered. A “canonical functions” approach is used. This approach replaces the integration of CE for the eigenvector with unknown initial values by the integration of CE for “canonical” functions having well-defined initial values at an arbitrary origin r0(0 < r0 < ∞). The terms of the reactance matrix are then deduced from these canonical functions. The results are independent from the choice of r0. It is shown that many conventional difficulties in the numerical application are avoided, mainly that of the choice of the starting boundary variable rs∼0 which is automatically determined. The results of the present method are compared to those of other methods and show excellent agreement with previous accurate results

The Canonical Function Method and its applications in quantum physics

“…The CFM developed initially by Kobeissi [35] and his coworkers to integrate the RSE, consists of writing the general solution y(r) representing the probability amplitude u l (E; r) as a function of the radial distance r in terms of two basis functions α(E; r) and β(E; r) for some energy E. Generally, the RSE is rewritten in a system of units such thath = 1, 2µ = 1 (see Appendix on units):…”

“…In order to gauge the accuracy of the spectra, we perform the integration of the RSE with two different methods: the fixed step RK4 and the variable step VSCA methods [35].…”

“…The integration of the RSE is performed with the VSCA method [35]. This method is superior to fixed step methods, and is based on a series expansion of the potential and the corresponding solution to an order such that a required tolerance criterion is met.…”

High-precision evaluation of the vibrational spectra of long-range molecules

“…During the integration, the ratio of the functions is monitored until saturation signaling the stability of the eigenvalue spectrum [8].…”

Distorted waves with exact nonlocal exchange: A canonical function approach

“…The aim of the present work is to develop an alternative numerical method, based on the Canonical Function approach (Kobeissi et al [10,11,12], which overcomes the problems of numerical stability. In this approach two independent sets of solutions are generated starting at some central point and integrating inwards to the origin and outwards to the asymptotic region.…”