Title: Spectral Theory of Infinite Quantum Graphs,
Abstract: We investigate quantum graphs with infinitely many vertices and edges without
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.