(untitled) (ҚазҰУ Хабаршысы. Математика, механика, информатика сериясы, 2022)
The equation of motion of a microparticle in various force fields is the Schr¨odinger wave equation. Many questions of quantum mechanics, in particular the thermal radiation of electromagnetic waves, lead to the problem of separability of singular differential operators. One such operator is the above Schr?dinger operator. In this paper, the named operator is studied by the methods of functional analysis. Found sufficient conditions for the existence of a solution and the separability of an operator in a Hilbert space. All theorems were originally proved for the model Sturm-Liouville equation and extended to a more general case. In §1-2, for the nonlinear Sturm-Liouville equation, sufficient conditions are found that ensure the existence of an estimate for coercivity, and estimates of weight norms are obtained for the first derivative of the solution. In Sections 3-4 the results of Sections 1-2 are generalized for the Schr¨odinger equation in the case m = 3.