REGULAR BOUNDARY VALUE PROBLEMS FOR A SINGULAR SECOND-ORDER EQUATION

Abstract

This paper is devoted to the study of regular boundary value problems for a singular second order differential equation with a degeneracy at one endpoint of the interval. Such equations arise naturally in various problems of mathematical physics and require a special analysis, since the singular behavior of the coefficient near the degenerate point prevents the direct application of standard methods from the classical theory of boundary value problems. We introduce the maximal and minimal operators associated with the given singular equation and identify the natural weighted boundary traces that arise in this context. We then study the corresponding singular Cauchy problem, establish its unique solvability in an appropriate weighted Sobolev space, and obtain an explicit representation of the inverse operator. This operator plays the role of a basic correct restriction and is used in the subsequent construction of general regular boundary conditions. Our main goal is to characterize all regular boundary value problems generated by the considered singular differential expression. The analysis is carried out within the framework of the theory of correct restrictions of maximal operators and regular extensions of minimal operators in a Hilbert space. Furthermore, the boundary criterion for integral operators obtained in [13] is adapted to the singular weighted setting. Using these approaches, we obtain the general form of boundary conditions that generate regular realizations of the considered singular equation.