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.

Dedicated to the memory of M. Z. Solomyak (16.05.1931 - 31.07.2016)

References in corpus (4)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• On the spectra of carbon nano-structures
• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• Explicit formulas for heat kernels on diamond fractals

Cited by in corpus (16)

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• Spectral Estimates for Infinite Quantum Graphs
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• Quantum graphs on radially symmetric antitrees
• A Glazman-Povzner-Wienholtz Theorem on graphs
• A note on the Gaffney Laplacian on infinite metric graphs
• Non-compact quantum graphs with summable matrix potentials
• Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
• Laplacians on infinite graphs: discrete vs continuous
• Dissipative extensions and port-Hamiltonian operators on networks
• Embedded trace operator for infinite metric trees
• Spectral Theory for Sturm-Liouville operators with measure potentials through Otelbaev's function
• Upper Eigenvalue Bounds for the Kirchhoff Laplacian on Embbeded Metric Graphs
• An existence theory for nonlinear equations on metric graphs via energy methods
• The $k$-adjacency operators and adjacency Jacobi matrix on distance-regular graphs
• Some spectral comparison results on infinite quantum graphs