Convergence of the Schwinger - DeWitt Expansion for Some Potentials

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.