On the Evolution Operator Kernel for the Coulomb and Coulomb–like Potentials

With a help of the Schwinger — DeWitt expansion analytical properties of the evolution operator kernel for the Schrödinger equation in time variable t are studied for the Coulomb and Coulomb-like (which behaves themselves as 1/|~q| when |~q| → 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 δ-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 δ-like singularity and for which the initial condition is fulfilled in rigorous sense (such as V (q) = − 2 1 cosh q for integer λ).