On Convergence of the Schwinger - DeWitt Expansion

TL;DR

This paper investigates the convergence properties of the Schwinger-DeWitt expansion in quantum mechanics, revealing conditions under which it converges for certain potentials and fixed parameters, challenging the usual assumption of divergence.

Contribution

It demonstrates that the Schwinger-DeWitt expansion can converge for specific potentials and fixed charge values, providing insights into the natural discreteness of charge.

Findings

Divergence occurs only when the coupling constant is variable.

The expansion converges for some potentials at discrete charge values.

Potential class reproduces charge discreteness naturally.

Abstract

The Schwinger - DeWitt expansion for the evolution operator kernel of the Schrodinger equation is studied for convergence. It is established that divergence of this expansion which is usually implied for all continuous potentials, excluding ones of the form V(q)=aq^2+bq+c, really takes place only if the coupling constant g is treated as independent variable. But the expansion may be convergent for some kinds of the potentials and for some discrete values of the charge, if the latter is considered as fixed parameter. Class of such potentials is interesting because inside of it the property of discreteness of the charge in the nature is reproduced in the theory in natural way.