Self-adjoint extensions and spectral analysis in the generalized Kratzer problem

TL;DR

This paper rigorously constructs self-adjoint operators and analyzes the spectral properties of the Kratzer potential in quantum mechanics, extending previous methods to a class of molecular interaction potentials.

Contribution

It provides a complete characterization of self-adjoint extensions and spectral solutions for the Kratzer potential, advancing the mathematical understanding of molecular quantum models.

Findings

Constructed all self-adjoint Schrödinger operators for the potential

Represented rigorous solutions to spectral problems

Extended previous methods to new potential class

Abstract

We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic motion of a particle in the potential field $V(x)=g_{1}x^{-1}+g_{2}x^{-2}$. For $g_{2}>0$ and $g_{1}<0$, the potential is known as the Kratzer potential and is usually used to describe molecular energy and structure, interactions between different molecules, and interactions between non-bonded atoms. We construct all self-adjoint Schrodinger operators with the potential $V(x)$ and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving spectral problems, we follow the Krein's method of guiding functionals. This work is a continuation of our previous works devoted to Coulomb, Calogero, and Aharonov-Bohm potentials.