Non-compact Quantum Graphs with Summable Matrix Potentials

Let G be a metric non-compact connected graph with finitely many edges. The main object of the paper is the Hamiltonian Hα associated in L2(G; Cm) with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian Hα as well as any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of Hα. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of Hα is obtained. Additionally, for a star graph G a formula is found for the scattering matrix of the pair { Hα, HD} , where HD is the Dirichlet operator on G. © 2020, Springer Nature Switzerland AG.