Particle creation by peak electric fields

There exist few configurations of t-electric potential steps and x-electric potential steps (i.e., electric fields that are switched on and off t → ±∞ and x → ±∞, respectively) that admit exact solutions of the Dirac equation and then detailed calculations of quantum processes in the presence of such external fields. We call them exactly solvable cases. Here we present new exactly solvable cases for such steps that have specific peak forms