Particle scattering and vacuum instability in a constant inhomogeneous electric field of particular peak configuration that consists of two (exponentially increasing and exponentially decreasing) independent parts are studied. It presents a new kind of external field where exact solutions of the Dirac and Klein-Gordon equations can be found. We obtain and analyze in-and out-solutions of the Dirac and Klein-Gordon equations in this configuration. By their help we calculate probabilities of particle scattering and characteristics of the vacuum instability. In particular, we consider in details three configurations: a smooth peak, a sharp peak, and a strongly asymmetric peak configuration.We find asymptotic expressions for total mean numbers of created particles and for vacuum-tovacuum transition probability. We discuss a new regularization of the Klein step by the sharp peak and compare this regularization with another one given by the Sauter potential.
There exists a clear physical motivation for theoretical studies of the vacuum instability related to the production of electron-positron pairs from a vacuum due to strong external electric fields. Various nonperturbative (with respect to the external fields) calculation methods were developed. Some of these methods are based on possible exact solutions of the Dirac equation. Unfortunately, there are only few cases when such solutions are known. Recently, an approximate but still nonperturbative approach to treat the vacuum instability caused by slowly varying t-electric potential steps (time dependent external fields that vanish as |t| → ∞), which does not depend on the existence of the corresponding exact solutions, was formulated in Ref. [S. P. Gavrilov, D. M. Gitman, Phys. Rev. D 95, 076013 (2017)]. Here, we present an approximate calculation method to treat nonperturbatively the vacuum instability in arbitrary weakly inhomogeneous x-electric potential steps (timeindependent electric fields of a constant direction that are concentrated in restricted space areas, which means that the fields vanish as |x| → ∞) in the absence of the corresponding exact solutions. Defining the weakly inhomogeneous regime in general terms, we demonstrate the universal character of the vacuum instability. This universality is associated with a large density of states excited from the vacuum by the electric field. Such a density appears in our approach as a large parameter. We derive universal representations for the total number and current density of the created particles. Relations of these representations with a locally constant field approximation for Schwinger's effective action are found.Center for Extreme Light Studies (XCELS), is slowly approaching the critical field strengths for observable pair production (see Ref.[13] for a review). Thus, there exists a clear physical motivation for theoretical studies of the vacuum instability. Firstly, it seems necessary to us to mention theoretical works devoted to various nonperturbative (with respect to the external field) calculation methods. Some of these methods are formulated for time-dependent external fields that vanish as |t| → ∞ (for t-electric potential steps in what follows) and are based on possible exact solutions of the Dirac equation; see, e.g., [6,[14][15][16][17]. Some of the methods are based on the analysis of the Schwinger effective action (see [18] for a review). The so-called derivative expansion approximation method, being applied to the Schwinger effective action, allows one to treat effectively arbitrary slowly varying in time strong fields [19,20]. We note that the locally constant field approximation (LCFA), which is to limit oneself to leading contributions of the derivative expansion of the effective action, allows for reliable results for electromagnetic fields of arbitrary strength; see, for example, Refs. [21,22]. An alternative approach to treat slowly varying t-electric potential steps, which does not depend on the existence of the corresponding exa...
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