  The Schwinger--DeWitt expansion for the evolution operator kernel is used to
investigate analytical properties of the Schr\"odinger equation solution in
time variable. It is shown, that this expansion, which is in general
asymptotic, converges for a number of potentials (widely used, in particular,
in one-dimensional many-body problems), and besides, the convergence takes
place only for definite discrete values of the coupling constant. For other
values of charge the divergent expansion determines the functions having
essential singularity at origin (beyond usual $\delta$-function). This does not
permit one to fulfil the initial condition. So, the function obtained from the
Schr\"odinger equation cannot be the evolution operator kernel. The latter,
rigorously speaking, does not exist in this case. Thus, the kernel exists only
for definite potentials, and moreover, at the considered examples the charge
may have only quantized values.
