A First-Order System of Equations on a Compact Star Graph

Abstract

This study concerns a boundary value problem generated by a system of first-order differential equations and some symmetric boundary conditions on a compact star graph. Unlike usual quantum graph Hamiltonians, the Hamiltonian considered in this paper acts on vector-valued functions living on the edges. Appropriate vertex conditions are introduced to ensure the corresponding boundary value problem is symmetric. In particular, coupling and separated conditions are imposed at the central and boundary vertices, respectively. Moreover, the general form of the vertex conditions at the central vertex is discussed. Finally, the characteristic function is constructed for the symmetric boundary value problem generated by the first-order system and specific coupling vertex conditions.