Potential splitting approach to multichannel Coulomb scattering: the driven Schr\"odinger equation formulation

TL;DR

This paper introduces a new method for multichannel Coulomb scattering that reformulates the Schrödinger equation into inhomogeneous equations with purely outgoing boundary conditions, improving computational efficiency especially with complex scaling techniques.

Contribution

It presents a novel potential splitting approach combined with integral equations, enabling Coulomb scattering problems to be solved more efficiently and accurately.

Findings

Reformulated scattering problem suitable for exterior complex scaling

Avoids Coulomb functions in calculations when using asymptotic expansion

Numerical results demonstrate the method's effectiveness on model problems

Abstract

In this paper we suggest a new approach for the multichannel Coulomb scattering problem. The Schr\"{o}dinger equation for the problem is reformulated in the form of a set of inhomogeneous equations with a finite-range driving term. The boundary conditions at infinity for this set of equations have been proven to be purely outgoing waves. The formulation {presented here} is based on splitting the interaction potential into a finite range core part and a long range tail part. The conventional matching procedure coupled with the integral Lippmann-Schwinger equations technique are used in the formal theoretical basis of this approach. The reformulated scattering problem is suitable for application in the exterior complex scaling technique: the practical advantage is that after the complex scaling the problem is reduced to a boundary problem with zero boundary conditions. The Coulomb wave functions are used only at a single point: if this point is chosen to be at a sufficiently large distance, on using the asymptotic expansion of Coulomb functions, one may completely avoid the Coulomb functions in the calculations. The theoretical results are illustrated with numerical calculations for two models.