Resonant states in an attractive one dimensional cusp potential

TL;DR

This paper analyzes the Dirac equation with a symmetric attractive cusp potential, revealing how bound states evolve into resonances as potential strength increases, and characterizing their properties.

Contribution

It provides an exact solution for the Dirac equation in a cusp potential and explores the transition from bound states to resonances, including lifetime calculations.

Findings

Bound states sink into the Dirac sea with increasing potential

Resonance states are characterized by phase shifts and Breit-Wigner relation

Limit case reduces to a delta point interaction

Abstract

We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetric attractive cusp potential. The components of the spinor solution are expressed in terms of Whittaker functions. We compute the bound states solutions and show that, as the potential amplitude increases, the lowest energy state sinks into the Dirac sea becoming a resonance. We characterize and compute the lifetime of the resonant state with the help of the phase shift and the Breit-Wigner relation. We discuss the limit when the cusp potential reduces to a delta point interaction.