Electron-positron pairs production in an electric potential of massive cores

Negative energy states of electrons bounded by a massive core with the charge-mass-radio Q/M and macroscopic radius R$_c$ are discussed. We show that the negative energies of bound states are lower than the negative electron mass-energy (-mc$^2$), and energy-level-crossing occurs. If these bound states are not occupied, electron-positron pair production takes place by quantum tunneling. Electrons fill into these bound states and positrons go to infinity. We explicitly calculate the rate of such pair-production, and compare it with the rates of electron-positron production by the Sauter-Euler-Heisenberg-Schwinger and Hawking processes.