Spectral properties of the Dirac equation in unbounded vector potentials

We study by numerical methods the Dirac equation in linear and quadratic potentials with pure vector coupling. We determine the spectral concentration of the continuous spectrum and we prove that it is well described by a sum of Breit–Wigner lines. The width of the line with lowest positive energy reproduces very well the Schwinger pair production rate. We then treat the quadratic potential using the methods of the perturbation theory. The problem is singular and the Distributional Borel Sum appears to be the most well suited tool to give answers and to describe the spectral properties of the system. The Padé approximants have been used for calculating the distributional Borel transform. A complete agreement between the two methods has been found.