Study of the convergence of the Schwinger–De Witt expansion for certain potentials

It is established that Schwinger–de Witt expansion is convergent for the potential $V=q^2/2+g/q^2$ (here $g=\lambda(\lambda-1)/2$ and $\lambda$ is integer number) and for a number of three-dimensional potentials with separated variables, but is divergent for the potentials $V=qe^{aq}$, $V=-ge^{-a^2q^2}$. Thereby it is shown that the initial condition for the evolution operator kernel for two latter potentials is fulfilled only in asymptotic sense. An outstanding role of the potentials for which Schwinger–de Witt expansion converges is discussed.