Variable-phase method and Levinson's theorem in two dimensions: Application to a screened Coulomb potential

TL;DR

This paper applies the variable-phase method to analyze scattering and bound states in a 2D screened Coulomb potential, revealing a new relationship between screening length and bound states, and exploring degeneracies as screening varies.

Contribution

It introduces a 2D formulation of Levinson's theorem and uncovers a simple relationship between screening length and the number of bound states.

Findings

A new 2D Levinson's theorem formulation

Discovery of a relationship between screening length and bound states

Degeneracy of bound states with increasing screening length

Abstract

The variable-phase approach is applied to scattering and bound states in an attractive Coulomb potential, statically screened by a two-dimensional (2D) electron gas. A 2D formulation of Levinson's theorem is used for bound-state counting and a hitherto undiscovered, simple relationship between the screening length and the number of bound states is found. As the screening length is increased, sets of bound states with differing quantum numbers appear degenerately.