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

The one-dimensional Schrödinger equation for a class of potentials $V(|x|)$ which vanish at infinity and present dominant singularity at the origin in the form $α/|x|^β$ ($0<β\leq 2$) 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 $β$, are discussed. Explicit solutions for the hydrogen atom and the Kratzer potential are presented.