Open AccessQuantum graphs with summable matrix potentials
Author(s) -
Ya. I. Granovskyi,
M. M. Malamud,
Hagen Neidhardt
Publication year - 2019
Publication title -
doklady akademii nauk. rossijskaâ akademiâ nauk
Language(s) - English
Resource type - Journals
ISSN - 0869-5652
DOI - 10.31857/s0869-565248815-10
Subject(s) - quantum graph , mathematics , hamiltonian (control theory) , boundary value problem , eigenvalues and eigenvectors , dirichlet boundary condition , absolute continuity , operator (biology) , pure mathematics , combinatorics , mathematical physics , graph , mathematical analysis , quantum mechanics , physics , mathematical optimization , biochemistry , chemistry , repressor , transcription factor , gene
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 Hα associated in L2(G; Cm) 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 Hα 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 Hα 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 Hα is obtained. Also we find a formula for the scattering matrix of the pair {Hα, HD} where HD is the operator of the Dirichlet problem on the graph.