Vacuum instability in a constant inhomogeneous electric field: a new example of exact nonperturbative calculations

Basic quantum processes (such as particle creation, reflection, and transmission on the corresponding Klein steps) caused by inverse-square electric fields are calculated. These results represent a new example of exact nonperturbative calculations in the framework of QED. The inverse-square electric field is time-independent, inhomogeneous in the x -direction, and is inversely proportional to x squared. We find exact solutions of the Dirac and Klein–Gordon equations with such a field and construct corresponding in- and out-states. With the help of these states and using the techniques developed in the framework of QED with x -electric potential steps, we calculate characteristics of the vacuum instability, such as differential and total mean numbers of particles created from the vacuum and vacuum-to-vacuum transition probabilities. We study the vacuum instability for two particular backgrounds: for fields widely stretches over the x -axis (small-gradient configuration) and for the fields sharply concentrates near the origin x = 0 (sharp-gradient configuration). We compare exact results with ones calculated numerically. Finally, we consider the electric field configuration, composed by inverse-square fields and by an x -independent electric field between them to study the role of growing and decaying processes in the vacuum instability.