Numerical Continuation of Bound and Resonant States of the Two Channel Schr\"odinger Equation

TL;DR

This paper extends numerical continuation methods to track resonant states in two-channel Schrödinger equations with energy separation, using uniformization to handle branch cuts, enabling detailed analysis of resonance paths.

Contribution

It introduces a novel approach combining regularization and uniformization to follow resonances in two-channel systems with energy separation, expanding previous methods.

Findings

Successfully extended the method to two-channel systems with energy separation.

Demonstrated the approach with extensive numerical examples.

Enabled detailed tracking of resonance paths in complex energy planes.

Abstract

Resonant solutions of the quantum Schr\"odinger equation occur at complex energies where the S-matrix becomes singular. Knowledge of such resonances is important in the study of the underlying physical system. Often the Schr\"odinger equation is dependent on some parameter and one is interested in following the path of the resonances in the complex energy plane as the parameter changes. This is particularly true in coupled channel systems where the resonant behavior is highly dependent on the strength of the channel coupling, the energy separation of the channels and other factors. In previous work it was shown that numerical continuation, a technique familiar in the study of dynamical systems, can be brought to bear on the problem of following the resonance path in one dimensional problems and multi-channel problems without energy separation between the channels. A regularization can be defined that eliminates coalescing poles and zeros that appear in the S-matrix at the origin due to symmetries. Following the zeros of this regularized function then traces the resonance path. In this work we show that this approach can be extended to channels with energy separation, albeit limited to two channels. The issue here is that the energy separation introduces branch cuts in the complex energy domain that need to be eliminated with a so-called uniformization. We demonstrate that the resulting approach is suitable for investigating resonances in two-channel systems and provide an extensive example.