Particle resonance in the Dirac equation in the presence of a delta interaction and a perturbative hyperbolic potential

TL;DR

This paper investigates how a weak hyperbolic tangent electric field influences the resonance behavior of the energy spectrum in a one-dimensional Dirac equation with a delta potential, revealing dependence on field strength and potential type.

Contribution

It provides an explicit solution using hypergeometric functions and derives an approximate resonance condition, comparing hyperbolic and linear perturbative potentials.

Findings

Resonant behavior depends on electric field strength at the delta interaction

Explicit solutions are obtained in terms of hypergeometric functions

Resonances characterized by phase shift and Wigner delay time

Abstract

In the present article we show that the energy spectrum of the one-dimensional Dirac equation, in the presence of an attractive vectorial delta potential, exhibits a resonant behavior when one includes an asymptotically spatially vanishing weak electric field associated with a hyperbolic tangent potential. We solve the Dirac equation in terms of Gauss hyper-geometric functions and show explicitly how the resonant behavior depends on the strength of the electric field evaluated at the support of the point interaction. We derive an approximate expression for the value of the resonances and compare the results calculated for the hyperbolic potential with those obtained for a linear perturbative potential. Finally, we characterize the resonances with the help of the phase shift and the Wigner delay time.