  The one-dimensional Schr\"{o}dinger equation for a class of potentials
$V(|x|)$ which vanish at infinity and present dominant singularity at the
origin in the form $\alpha /|x|^{\beta}$ ($0<\beta \leq 2$) is investigated.
The Hermiticity of the operators related to observable physical quantities is
used to determinate the proper boundary conditions. Double degeneracy and
exclusion of symmetric solutions, consonant the value of $\beta $, are
discussed. Explicit solutions for the hydrogen atom and the Kratzer potential
are presented.
