On quantum graphs

Quantum graphs with infinitely many vertices will be discussed. The main novelty is that we don't assume the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types. The talk is based on the following papers: 1. Kostenko, A., Malamud, M.: 1-D Schrodinger operators with local point interactions on a discrete set. JDE, v. 249, 253–304 (2010). 2. P. Exner, A. Kostenko, M.M. Malamud, and H. Neidhardt, {Spectral Theory of Infinite Quantum Graphs}, Annales Henri Poincare, V. 19, No 11, (2018), p. 3457 – 3510.