On Molchanov's Condition for the Spectrum Discreteness of a Quantum Graph Hamiltonian with δ-Coupling

The spectrum discreteness Molchanov's condition for a potential on the real axis (or half-axis) is well known. The goal of this work is to obtain an analogous condition for a quantum graph. We deal with δ-type conditions at the graph vertices. The Schrödinger operator with a potential is considered at each edge. It is assumed that the graph has infinite leads (edges) or/and infinite chains of vertices such that the neighbouring vertices are connected by finite number of edges. Molchanov-type theorem for the quantum graph Hamiltonian spectrum discreteness is proved.