Perturbation theory for metastable states of the Dirac equation with quadratic vector interaction

The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The cases in one spatial dimension and in three spatial dimensions with spherical symmetry are explicitly studied. 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 Padé approximants for calculating the distributional Borel transform. A complete agreement is found with previous nonperturbative 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.