States of the Dirac Equation in Confining Potentials

We study the Dirac equation in confining potentials with pure vector coupling, proving the existence of metastable states with longer and longer lifetimes as the nonrelativistic limit is approached and eventually merging with continuity into the Schrödinger bound states. The existence of these states could concern high energy models and possible resonant scattering effects in systems like graphene. We present numerical results for the linear and the harmonic cases and we show that the density of the states of the continuous spectrum is well described by a sum of Breit-Wigner lines. The width of the line with lowest positive energy well reproduces the Schwinger pair production rate for a linear potential: this gives an explanation of the Klein paradox for bound states and a new concrete way to get information on pair production in unbounded, nonuniform electric fields, where very little is known.

Figure 1

The density of the states for the linear potential with $\Omega =0.3$, 1. The scale for $\Omega =0.3$ must be multiplied by ${10}^2$.

Figure 2

$\rho (\lambda ,\Omega )$ for the quadratic potential.

Figure 3

The first BW maxima vs $\Omega$ for the quadratic potential. Nonrelativistic bound states correspond to odd integers.

Figure 4

Left plot: the width of the first resonance (diamonds) compared with the pair production curve $w^f(\Omega )$ (solid line). Right plot: the width of the first resonance (circles) for a quadratic potential. The solid line, giving a very good fit, is $w(\Omega )=-{\pi }^{-1}\ln [1-\exp \{-\pi /(4^2{\Omega }^{3/2})\}]$, analogous to $w^f(\Omega )$.