The Dyson instability and asymptotics of the perturbation series in QED

The probability of the ground state decay at e 2 < 0 is calculated by the steepest descent method in fermionic electrodynamics. The saddle points are the solution of some equations which have been obtained by calculating the asymptotic of the Dirac operator determinant in a very strong external field. The SO(3) × O(2) solutions are found explicitly. The main contribution from them to the N th coefficient of perturbation theory is proportional to (−α) N 2 Γ( N 2 ) , where S = 2 2 3 −3 2 π 3 .