Effective Action of QED in Electric Field Backgrounds II: Spatially Localized Fields

We find the Bogoliubov coefficient from the tunneling boundary condition on a charged particle coupled to a static electric field $E_0 sech^2 (z/L)$ and, using the regularization scheme in Phys. Rev. D 78, 105013 (2008), obtain the exact one-loop effective action in scalar and spinor QED. It is shown that the effective action satisfies the general relation between the vacuum persistence and the mean number of produced pairs. We advance an approximation method for general electric fields and show the duality between the space-dependent and time-dependent electric fields of the same form at the leading order of the effective actions.