  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 $\lambda$).
