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 electromagneticwaves, lead to the problem of separability of singular differential operators. One such operator isthe above Schr?dinger operator. In this paper, the named operator is studied by the methods offunctional analysis. Found sufficient conditions for the existence of a solution and the separabilityof an operator in a Hilbert space. All theorems were originally proved for the model Sturm-Liouvilleequation and extended to a more general case.In §1-2, for the nonlinear Sturm-Liouville equation, sufficient conditions are found that ensurethe existence of an estimate for coercivity, and estimates of weight norms are obtained for thefirst derivative of the solution. In Sections 3-4 the results of Sections 1-2 are generalized for theSchr¨odinger equation in the case m = 3.

Keywords

• nonlinear equations
• continuous operator
• equivalence
• potential function