Resonances in the one-dimensional Dirac equation in the presence of a point interaction and a constant electric field

TL;DR

This paper investigates how a weak electric field influences the resonance behavior of the energy spectrum in a one-dimensional Dirac equation with a point interaction, providing explicit solutions and approximate resonance energies.

Contribution

It introduces an analytical approach to understanding resonances in the Dirac equation with point interactions under electric fields, including explicit solutions and resonance energy approximations.

Findings

Resonant behavior appears in the energy spectrum with an electric field.

Explicit solutions involve parabolic cylinder functions.

Approximate resonance energies depend on electric field and interaction strength.

Abstract

We show that the energy spectrum of the one-dimensional Dirac equation in the presence of a spatial confining point interaction exhibits a resonant behavior when one includes a weak electric field. After solving the Dirac equation in terms of parabolic cylinder functions and showing explicitly how the resonant behavior depends on the sign and strength of the electric field, we derive an approximate expression for the value of the resonance energy in terms of the electric field and delta interaction strength.