Applying numerical continuation to the parameter dependence of solutions of the Schr\"odinger equation

TL;DR

This paper introduces a numerical continuation method based on bifurcation theory to analyze how bound states in the Schrödinger equation transition into resonances as system parameters change, improving prediction accuracy.

Contribution

It develops a robust continuation technique using Keller's Pseudo-Arclength method to study bound state to resonance transitions in the Schrödinger equation, addressing challenges with non-smooth S-matrix functions.

Findings

Successfully applied to model radial Schrödinger problems.

Enhanced understanding of resonance formation in molecular systems.

Demonstrated robustness and efficiency of the method.

Abstract

In molecular reactions at the microscopic level the appearance of resonances has an important influence on the reactivity. It is important to predict when a bound state transitions into a resonance and how these transitions depend on various system parameters such as internuclear distances. The dynamics of such systems are described by the time-independent Schr\"odinger equation and the resonances are modeled by poles of the S-matrix. Using numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, we are able to develop efficient and robust methods to study the transitions of bound states into resonances. By applying Keller's Pseudo-Arclength continuation, we can minimize the numerical complexity of our algorithm. As continuation methods generally assume smooth and well-behaving functions and the S-matrix is neither, special care has been taken to ensure accurate results. We have successfully applied our approach in a number of model problems involving the radial Schr\"odinger equation.