Degenerate Boundary Conditions on a Geometric Graph

Abstract

The boundary conditions of the Sturm–Liouville problem defined on a star-shaped geometric graph of three edges are studied. It is shown that, if the lengths of the edges are different, then the Sturm–Liouville problem does not have degenerate boundary conditions. If the lengths of the edges and the potentials are identical, then the characteristic determinant of the Sturm–Liouville problem cannot be equal to a constant different from zero. However, the set of Sturm–Liouville problems for which the characteristic determinant is identically zero is infinite (continuum). In this way, in contrast to the Sturm–Liouville problem defined on an interval, the set of boundary value problems on a star-shaped graph whose spectrum completely fills the entire plane is much richer. In the particular case when the minor ${{A}_{{124}}}$ of the coefficient matrix is nonzero, this set consists of not two problems, as in the case of the Sturm–Liouville problem given on an interval, but rather of 18 classes, each containing two to four arbitrary constants.