Numerical Continuation of resonances and bound states in coupled channel Schr\"odinger equations

TL;DR

This paper presents a novel numerical continuation method for tracking bound and resonant states in coupled channel Schrödinger equations, utilizing S-matrix regularization to automate parameter dependence analysis.

Contribution

It introduces a regularization technique transforming the S-matrix for improved smoothness, enabling automated continuation of quantum states in coupled systems.

Findings

Successfully applied to model problems

Enhanced ability to track states across parameter ranges

Improved numerical stability and accuracy

Abstract

In this contribution, we introduce numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schr\"odinger equation. We extend previous work on the subject to systems of coupled equations. Bound and resonant states of the Schr\"odinger equation can be determined through the poles of the S-matrix, a quantity that can be derived from the asymptotic form of the wave function. We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties, thus making it amenable to numerical continuation. This allows us to automate the process of tracking bound and resonant states when parameters in the Schr\"odinger equation are varied. We have applied this approach to a number of model problems with satisfying results.