Perturbation Theory for Metastable States of the Dirac Equation with Quadratic Vector Interaction

TL;DR

This paper applies advanced perturbation methods, including Distributional Borel Sum, to analyze the spectral properties of the Dirac equation with quadratic vector potential, providing numerical results that align with non-perturbative solutions.

Contribution

It introduces a perturbation approach using Distributional Borel Sum for the Dirac equation with quadratic vector interaction, addressing the divergence of the series.

Findings

Perturbation series is asymptotic and requires special summation techniques.

Numerical results agree with non-perturbative solutions.

Distributional Borel Sum effectively describes spectral properties.

Abstract

The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The problem is singular and the perturbation series is asymptotic, so that the methods for dealing with divergent series must be used. Among these, the Distributional Borel Sum appears to be the most well suited tool to give answers and to describe the spectral properties of the system. A detailed investigation is made in one and in three space dimensions with a central potential. We present numerical results for the Dirac equation in one space dimension: these are obtained by determining the perturbation expansion and using the Pad\'e approximants for calculating the distributional Borel transform. A complete agreement is found with previous non-perturbative results obtained by the numerical solution of the singular boundary value problem and the determination of the density of the states from the continuous spectrum.