Convergence of the Schwinger - DeWitt Expansion for Some Potentials

TL;DR

This paper investigates the convergence properties of the Schwinger-DeWitt expansion for certain potentials in the Schrödinger equation, identifying specific 'good' potentials where the expansion converges and charge quantization occurs.

Contribution

It identifies and analyzes potentials for which the Schwinger-DeWitt expansion converges, revealing discrete charge values and their implications for divergence-free quantum theories.

Findings

Convergence occurs for potentials V=g/x^2, V=-g/cosh^2 x, V=g/sinh^2 x, V=g/sin^2 x when g=λ(λ-1)/2 with integer λ.

These 'good' potentials have no divergences and exhibit charge quantization.

Convergence is restricted to discrete charge values, indicating quantized theories.

Abstract

It is studied time dependence of the evolution operator kernel for the Schr\"odinger equation with a help of the Schwinger -- DeWitt expansion. For many of potentials this expansion is divergent. But there were established nontrivial potentials for which the Schwinger -- DeWitt expansion is convergent. These are, e.g., V=g/x^2, V=-g/cosh^2 x, V=g/sinh^2 x, V=g/sin^2 x. For all of them the expansion is convergent when $g=\lambda (\lambda -1)/2$ and $\lambda$ is integer. The theories with these potentials have no divergences and in this meaning they are "good" potentials contrary to other ones. So, it seems natural to pay special attention namely to these "good" potentials. Besides convergence they have other interesting feature: convergence takes place only for discrete values of the charge $g$. Hence, in the theories of this class the charge is quantized.