On Electron-Positron Pair Production by a Spatially Nonuniform Electric   Field
  A detailed analysis of electron-positron pair creation induced by a spatially nonuniform and static electric field from vacuum is presented. A typical example is provided by the Sauter potential. For this potential, we derive the analytic expressions for vacuum decay and pair production rate accounted for the entire range of spatial variations. In the limit of a sharp step, we recover the divergent result due to the singular electric field at the origin. The limit of a constant field reproduces the classical result of Euler, Heisenberg and Schwinger, if the latter is properly averaged over the width of a spatial variation. The pair production by the Sauter potential is described for different physical regimes from weak to strong fields. In all these regimes, the locally constant-field rate is shown to be the upper limit. 
A. Chervyakov and H. Kleinert
hep-th