Mutual transformation among bound, virtual and resonance states in one-dimensional rectangular potentials

TL;DR

This paper explores how bound, virtual, and resonance states in a one-dimensional rectangular potential transform into each other by analyzing the pole spectrum of the S-matrix as the potential is complexified, revealing global spectral features.

Contribution

It introduces a global analysis of the pole spectrum trajectories in the complex momentum plane using a complex extension of the potential, complementing previous local studies.

Findings

Pole trajectories reveal mutual transformations among states.

Complex extension of potential elucidates spectral structures.

Provides a global perspective on state transitions.

Abstract

A detailed analysis has been made by R.Zavin and N.Moiseyev(2004 J. Phys. A: Math, Gen, \textbf{37} 4619) for the change of bound states into resonance states via coalescence of virtual states in a one-dimensional symmetric rectangular attractive potential as it becomes shallow, with convergent wave functions of virtual and resonance states by the complex scaling method. As a complement to such an analysis, we discuss some global features of the pole spectrum of the S-matrix by using a complex extension of the real potential $V^{(\mathrm{real})}$ to $e^{i\alpha}V^{(\mathrm{real})}$ with a real phase $\alpha$. We show the structures of trajectories of poles developed for the change of $\alpha$ in the complex momentum plane, which is useful to understand the mutual transformation among the bound, virtual and resonance states.