Schwinger mechanism in dS₂ and AdS₂ revisited

Recalculating the Bogoliubov coefficients from the solutions in Phys. Rev. D 78, 103517 (2008), we obtain the mean number of boson pairs in a uniform electric field in the global coordinates dS_2 and AdS_2, which have the correct zero-field and zero-curvature limits, and study the vacuum persistence at one-loop. The mean number in AdS_2 gives the lowest limit to the Breitenloher-Freedman bound in the uniform electric field, and the mean numbers in dS_2 and AdS_2 satisfy the reciprocal relation N_{dS} (R, E) N_{AdS} (R, E) = 1 under the analytical continuation of the scalar curvature R.