We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic motion of a particle in the potential fieldFor g2 > 0 and g1 < 0, the potential is known as the Kratzer potential VK (x) and is usually used to describe molecular energy and structure, interactions between different molecules, and interactions between nonbonded atoms.We construct all self-adjoint Schrödinger 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.