On the Evolution Operator Kernel for the Coulomb and Coulomb--Like Potentials

TL;DR

This paper investigates the analytical properties of the evolution operator kernel for Coulomb and Coulomb-like potentials, revealing divergence in the Schwinger-DeWitt expansion and additional singularities at t=0, affecting the initial condition fulfillment.

Contribution

It demonstrates that Coulomb and Coulomb-like potentials have divergent Schwinger-DeWitt expansions and non-δ-like singularities at t=0, distinguishing them from certain well-behaved potentials.

Findings

Schwinger-DeWitt expansion diverges for Coulomb potentials

Kernels exhibit additional singularity at t=0 beyond δ-function

Initial conditions are fulfilled only asymptotically for these potentials

Abstract

With a help of the Schwinger --- DeWitt expansion analytical properties of the evolution operator kernel for the Schr\"odinger equation in time variable $t$ are studied for the Coulomb and Coulomb-like (which behaves themselves as $1/|\vec q|$ when $|\vec q| \to 0$) potentials. It turned out to be that the Schwinger --- DeWitt expansion for them is divergent. So, the kernels for these potentials have additional (beyond $\delta$-like) singularity at $t=0$. Hence, the initial condition is fulfilled only in asymptotic sense. It is established that the potentials considered do not belong to the class of potentials, which have at $t=0$ exactly $\delta$-like singularity and for which the initial condition is fulfilled in rigorous sense (such as $V(q) = -\frac{\lambda (\lambda-1)}{2} \frac {1}{\cosh^2 q}$ for integer $\lambda$).