SMOOTHNESS OF SOLUTIONS (SEPARABILITY) OF THE NONLINEAR STATIONARY SCHR¨ODINGER EQUATION

Abstract

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.