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We find the Bogoliubov coefficient from the tunneling boundary condition on acharged 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 theexact one-loop effective action in scalar and spinor QED. It is shown that theeffective action satisfies the general relation between the vacuum persistenceand the mean number of produced pairs. We advance an approximation method forgeneral electric fields and show the duality between the space-dependent andtime-dependent electric fields of the same form at the leading order of theeffective actions.

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