The one-dimensional Schrodinger 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 < 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, depending on the value of β, are discussed. Explicit solutions for the hydrogen atom and the Kratzer potential are presented.
singular potential; degeneracy; one-dimensional hydrogen atom; Kratzer potential; collapse to the center
