An elementary introduction to quantum graphs

Abstract

We describe some basic tools in the spectral theory of Schrödinger operator on metric graphs (also known as "quantum graph") by studying in detail some basic examples. The exposition is kept as elementary and accessible as possible. In the later sections we apply these tools to prove some results on the count of zeros of the eigenfunctions of quantum graphs.

31 pages, 17 figures

References in corpus (14)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Chaotic Scattering on Graphs
• The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums
• On the Spectral Gap of a Quantum Graph
• Topological Resonances in Scattering on Networks (Graphs)
• Nodal count of graph eigenfunctions via magnetic perturbation
• Magnetic interpretation of the nodal defect on graphs
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Universality of the momentum band density of periodic networks
• Some planar isospectral domains
• Non-Weyl resonance asymptotics for quantum graphs in a magnetic field
• Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs
• Eigenvalue estimates on quantum graphs
• Non-Weyl Resonance Asymptotics for Quantum Graphs

Cited by in corpus (5)

• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• A spectral duality in graphs and microwave networks
• Zero Measure and Singular Continuous Spectra for Quantum Graphs
• Spectral ratios and gaps for Steklov eigenvalues of balls with revolution-type metrics
• Topological Tenfold Classification and the Entanglement of Two Qubits