Schrödinger wave equation is known as an equation that governs the motion of a microparticle in various force fields. The problem of singular differential operators’ separability occurs due to examination of various issues in quantum mechanics, especially, thermal radiation of electromagnetic waves. One such operator is the above Schrödinger operator. In this paper, we study the specified operator using functional-analytic methods. We found sufficient conditions for existence of a solution and separability of an operator in the Hilbert space. All theorems were originally proved for the model Sturm–Liouville equation and extended to a more general case.

In the sections of the existence of the solution and the smoothness of the solution, sufficient conditions were found to ensure the existence of a coercivity estimate for the nonlinear Sturm-Liouville equation, and estimates of weight norms for the first derivative of the solution were obtained. The results of the sections the existence of the solution in the last sections and the smoothness of the solution are generalized for the Schrodinger equation in the case m=3.