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To find the pair production, absorption cross section and quasi-normal modesin background fields, we advance the monodromy method that makes use of theregular singular points of wave equations. We find the mean number of pairsproduced in background fields whose mode equations belong to the Riemanndifferential equation and apply the method to the three particular cases: (i)charges near the horizon of near-extremal black holes, (ii) charges withminimal energy under the static balance in nonextremal charged black holes, and(iii) charges in the Sauter-type electric fields. We then compare the resultsfrom the monodromy with those from the exact wave functions in terms of thehypergeometric functions with three regular singular points. The explicitelaboration of monodromy and the model calculations worked out here seem toreveal evidences that the monodromy may provide a practical technique to studythe spontaneous pair production in general black holes and electromagneticfields.

Monodromy Approach to Pair Production of Charged Black Holes and Electric Fields