Degenerate Boundary Conditions for the Sturm-Liouville Problem on a Geometric Graph

Abstract

We study the boundary conditions of the Sturm-Liouville problem posed on a star-shaped geometric graph consisting of three edges with a common vertex. We show that the Sturm-Liouville problem has no degenerate boundary conditions in the case of pairwise distinct edge lengths. However, if the edge lengths coincide and all potentials are the same, then the characteristic determinant of the Sturm-Liouville problem cannot be a nonzero constant and the set of Sturm-Liouville problems whose characteristic determinant is identically zero and whose spectrum accordingly coincides with the entire plane is infinite (a continuum). It is shown that, for one special case of the boundary conditions, this set consists of eighteen classes, each having from two to four arbitrary constants, rather than of two problems as in the case of the Sturm-Liouville problem on an interval.