A brief discussion on the possible bound states for a class of singular potentials

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.