Stable perturbations of boundary problems for di˙erential equations

Abstract

In this paper, we expand the class of nondegenerate two-point boundary value problems for the Sturm-Liouville equation, which have a complete system of eigenfunctions and associated functions in special function spaces. Such spaces depend on the length of support of the potential of the Sturm-Liouville equation. The formulated results clarify well-known results of V.A. Marchenko. Two-point boundary value problems for the Sturm-Liouville equation are divided into degenerate and nondegenerate boundary condi-tions in the sense of V.A. Marchenko. The main result of V.A. Marchenko asserts that systems of eigen-functions and associated functions of nondegenerate boundary value problems for the Sturm-Liouville equation form a complete system of functions in the space of square-summable functions. In this paper, the result of V.A. Marchenko is clarified in the following direction. There are operators with a complete system of eigenfunctions and associated functions in the space of square-summable functions among the degenerate boundary value problems in the sense of V.A. Marchenko. The presence of the completeness property depends on the length of support of the measure which is antisymmetry to the potential of the Sturm-Liouville equation.