Regularization, renormalization and consistency conditions in QED with x-electric potential steps

The present article is an important addition to the nonperturbative formulation of QED with x-steps presented by Gavrilov and Gitman in Phys. Rev. D. 93, 045002 (2016). Here we propose a new renormalization and volume regularization procedures which allow one to calculate and distinguish physical parts of different matrix elements of operators of the current and of the energy-momentum tensor, at the same time relating the latter quantities with characteristics of the vacuum instability. For this purpose, a modified inner product and a parameter {\tau} of the regularization are introduced. The latter parameter can be fixed using physical considerations. In the Klein zone this parameter can be interpreted as the time of the observation of the pair production effect. In the refined formulation of QED with x-steps, we succeeded to consider the backreaction problem. In the case of an uniform electric field E confined between two capacitor plates separated by a finite distance L, we see that the smallness of the backreaction implies a restriction (the consistency condition) on the product EL from above.