Phase shifts and resonances in the Dirac equation

TL;DR

This paper compares non-relativistic and relativistic scattering, revealing that relativistic effects introduce s-wave resonances absent in the non-relativistic case, and presents a numerical method for phase shift analysis.

Contribution

It provides a detailed analysis of phase shifts and resonances in the Dirac equation, highlighting differences from non-relativistic scattering and introducing a numerical phase shift extraction method.

Findings

Resonances occur for p-waves in non-relativistic scattering but also for s-waves in relativistic scattering.

Relativistic s-wave resonances happen when the potential is near supporting zero-energy solutions.

A numerical procedure for phase shift extraction is demonstrated with Gaussian potentials.

Abstract

We review the analytic results for the phase shifts delta_{l}(k) in non-relativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the non-relativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E = m. Resonances resulting from scattering from a barrier can be explained in terms of the `crossing' theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extract phase shifts for general short range potentials is introduced and illustrated by considering relativistic scattering from a Gaussian potential well and barrier.