  In non-relativistic quantum mechanics we study the Coulomb systems of
infinitely massive center of charge Z and two-three electrons: $(Z,e,e)$ and
$(Z,e,e,e)$. It is shown that in both cases the total energy curve in $Z$ is
smooth, without any visible irregularities. Thus, for both systems the physical
integer charges $Z=1,2,...$ do not play a distinguished role as would be
associated with charge quantization. By definition, a critical charge $Z_{cr}$
is a charge which separates a domain of the existence of bound states from a
domain of unbound ones (continuum). For both systems the critical charges are
found, $Z_{cr,2e}=0.91085$ and $Z_{cr,3e}=2.009$, respectively. Based on
numerical analysis, the Puiseux expansion in fractional powers of $(Z-Z_{cr})$
is constructed for both systems. Our results indicate the existence of a
square-root branch point singularity at $Z_{cr}$ with exponent 3/2. A
connection between the critical charge and the radius of convergence of
1/Z-expansion is briefly discussed.
