Quantum Graphs with Summable Matrix Potentials

Abstract

Let \(\mathcal{G}\) be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume that the length of at least one of the edges is infinite. The main object of this paper is the Hamiltonian \({{{\mathbf{H}}}_{\alpha }}\) associated in \({{L}^{2}}(\mathcal{G};{{\mathbb{C}}^{m}})\) 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 \({{{\mathbf{H}}}_{\alpha }}\) and any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring that the positive part of \({\mathbf{H}_{\alpha }}\) is purely absolutely continuous. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of the operator \({{{\mathbf{H}}}_{\alpha }}\) is obtained. Additionally, a formula is found for the scattering matrix of the pair \(\{ {{{\mathbf{H}}}_{\alpha }},{{{\mathbf{H}}}_{D}}\} \), where \({{{\mathbf{H}}}_{D}}\) is the operator of the Dirichlet problem on the graph.