Potentials with convergent Schwinger-DeWitt expansion

Convergence of the Schwinger-DeWitt expansion for the evolution operator kernel for special class of potentials is studied. It is shown that this expansion, which is in the general case asymptotic, converges for the potentials considered (widely used, in particular, in one-dimensional many-body problems), and that convergence takes place only for definite discrete values of the coupling constant. For other values of the charge, a divergent expansion determines the kernels having essential singularity at the origin (beyond the usual delta-function). If one considers only this class of potentials, then one can avoid many problems connected with asymptotic expansions, and one gets a theory with discrete values of the coupling constant that is in correspondence with the discreteness of charge in nature. This approach can be applied to quantum field theory.