On Convergence of the Schwinger–De Witt Expansion

Abstract

The Schwinger–De Witt expansion for the evolution operator kernel of the Schrödinger equation is studied for convergence. It is established that divergence of this expansion which is usually implied for all continuous potentials, excluding those of the form V(q)=aq2+bq+c, takes place only if the coupling constant g is treated as an independent variable. But the expansion may be convergent for some kinds of potentials and for some discrete values of charge, if the latter is considered as fixed parameter. Class of such potentials is interesting because inside it the property of discreteness of the charge in nature is reproduced in theory in the natural way.