Equilateral quantum graphs and boundary triples

Abstract

The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural generalisation of the discrete Laplace operator on the underlying graph. These generalised Laplacians are necessary in order to cover general vertex boundary conditions on the metric graph. In case of the standard (also named ``Kirchhoff'') boundary conditions, the discrete operator is the usual combinatorial Laplacian.

19 pages, some references added

References in corpus (6)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Spectral convergence of non-compact quasi-one-dimensional spaces
• Spectra of Schroedinger operators on equilateral quantum graphs
• Spectral Statistics for the Dirac Operator on Graphs
• Spectral analysis of metric graphs and related spaces
• First order approach and index theorems for discrete and metric graphs

Cited by in corpus (12)

• Spectral analysis on infinite Sierpinski fractafolds
• Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures
• On the nonlinear Dirac equation on noncompact metric graphs
• An example of unitary equivalence between self-adjoint extensions and their parameters
• Isospectral graphs with identical nodal counts
• Continuity of the integrated density of states on random length metric graphs
• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs
• Spectral analysis of metric graphs and related spaces
• Localization on quantum graphs with random edge lengths
• Eigenvalue bracketing for discrete and metric graphs
• Abstract graph-like space and vector-valued metric graphs