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

TL;DR

This paper investigates bound states in a class of singular potentials in one-dimensional quantum mechanics, analyzing boundary conditions, degeneracy, and providing explicit solutions for specific potentials like hydrogen and Kratzer.

Contribution

It offers a detailed analysis of boundary conditions and degeneracy for singular potentials, including explicit solutions for hydrogen and Kratzer potentials, expanding understanding of such systems.

Findings

Proper boundary conditions depend on the singularity strength.

Double degeneracy occurs for certain potentials.

Explicit solutions for hydrogen and Kratzer potentials are provided.

Abstract

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.