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.
Categories: math-ph math.MP quant-ph