Quantum graphs with summable matrix potentials

Let G be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume also that the length at least of one of the edges is infinite. The main object of the paper is Hamiltonian H_α associated in L²(G; C_m) with matrix Sturm-Liouville’s expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying technique of boundary triplets and the corresponding Weyl functions we show that the singular continuous spectrum of the Hamiltonian H_α as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring the positive part of the Hamiltonian H_α to be purely absolutely continuous. Under an additional condition on the potential matrix the Bargmann type estimate for the number of the negative eigenvalues of the operator H_α is obtained. Also we find a formula for the scattering matrix of the pair {H_α, H_D} where H_D is the operator of the Dirichlet problem on the graph.