Degenerate Sturm-Liouville Problem for Second-Order Differential Operators on Star-Graph

Abstract

In this paper, we present a comprehensive study of second-order differential operators on a star-graph geometric graph considering a star graph with three edges and a common vertex. We investigate the Dirichlet problem for a Sturm-Liouville operator defined on this network-type manifold. The Sturm-Liouville problem is formulated as a system of ordinary differential equations (1) on the individual edges, subject to the boundary conditions (2) and (3) at the common vertex. We assume that the condition holds, ensure the non-degeneracy of the boundary conditions by using a synthetic approach. We fully describe and solve the Dirichlet problem for the given second-order differential operator on the star graph. The key results include the characterization of the spectral parameter  the construction of the matrix A composed of the boundary condition coefficients, and the analysis of the minors of A. The findings of this work contribute to the understanding of second-order differential operators on network-type manifolds and provide a framework for addressing similar problems on more complex geometric graphs. The insights gained from this study have potential applications in various fields, such as: quantum mechanics, control theory, and network analysis.