Studying resonance in the complex charge plane

TL;DR

This paper introduces a novel indirect method to study resonances in the complex charge (Z) plane using complex scaling, mapping bound states and resonances onto poles of a resolvent operator, with demonstrated stability and comparison to existing results.

Contribution

It presents a new formulation for resonance analysis in the complex charge plane employing complex scaling, complementing existing methods in energy and angular momentum planes.

Findings

Resonances are mapped onto poles of a resolvent operator on the real Z-plane.

The method's stability is demonstrated against computational parameter variations.

Calculated resonances agree with existing results for a test potential.

Abstract

Potential resonances are usually investigated either directly in the complex energy plane or indirectly in the complex angular momentum plane. Another formulation complementing these two is presented in this work. It is an indirect method which studies resonances in a complex charge plane (Z-plane). The complex scaling (rotation) method is employed in the development of this formulation. Bound states spectrum and resonance energies are mapped onto a discrete set of poles of a resolvent operator on the real line of the Z-plane. These poles move along trajectories as we vary the energy. A finite L2 basis is used in the numerical implementation of the method. Stability of poles and trajectories against variations in all computational parameters is demonstrated. Resonances for a given potential example are calculated and compared with those obtained elsewhere. In this presentation, modest effort will be devoted to mathematical rigor.