}{\alpha }}\frac{\left( m_{e}^{2}+k^{2}\right) ^{2}}{\omega \sqrt{\frac{\omega ^{2}}{2}+\pi ^{2}T^{2}}}\times
\nonumber\\
&&\left[ \frac{1}{\omega ^{2}}+\frac{1}{4}\frac{1}{\frac{\omega
^{2}}{2}+\pi ^{2}T^{2}}+\frac{3}{8}\frac{\omega ^{2}}{\left(
\frac{\omega ^{2}}{2}+\pi ^{2}T^{2}\right) ^{2}}\right] ,\end{aligned}$$ giving $$\dot{n}_{\pm }\left( z,T\right) \approx \frac{3}{8\pi
^{2}}e^{2}E^{2}\left(
z,T\right) \exp \left[ \frac{25\pi }{48\lambda \left( z,T\right) }\right] \left[ 3\lambda \left( z,T\right) \right] ^{-1/2}\left[ 1-{\rm
erf} \left( \chi \left( z,T\right) \right) \right] ,$$ where $\lambda (z,T)=eE\left[ 1/V^{2}+\left( 1/3\pi \right)
V^{2}/E^{2}+\left( 2/3\pi ^{2}\right) V^{6}/E^{4}\right] $, $\chi =\sqrt{\lambda (z,T)/3\pi }E_{cr}/E+5\sqrt{\pi }/4\sqrt{3}\lambda \left(
z,T\right) $ and ${\rm erf}\left( x\right) =\left( 2/\sqrt{\pi
}\right) \int_{0}^{x}\exp \left( -u^{2}\right) du$.

Boundary effects in pair production in the electrosphere 
=========================================================

The electron-positron pair creation in the electrosphere takes place very close to the surface of the quark star, which represents the boundary of the system. Therefore electron-positron pair creation is localized to the half-space $z\geq 0$, and the pair creation takes place in electric field localized to a bounded region in the space. Boundary effects in electron-positron pair creation by electric fields confined in a finite region of space have been previously investigated, with the general result that finite-size effects induce large deviations of the production rate from what one deduces from the Schwinger formula [@Wa88; @Ma88; @Ma89].

In analyzing the surface effects on pair production there are two very different situations, 