Spectral Theory of Infinite Quantum Graphs

Spectral Theory of Infinite Quantum Graphs

We investigate spectral properties of quantum graphs with infinitely many 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 with Kirchhoff or, more generally, δ-type couplings at vertices and the corresponding...

Combinatorial Identities from the Spectral Theory of Quantum Graphs

We present a few combinatorial identities which were encountered in our work on the spectral theory of quantum graphs. They establish a new connection between the theory of random matrix ensembles and combinatorics.

The Spectral Radius of Infinite Graphs

Recently some important results have been proved showing that the gap between the largest eigenvalue A: of a finite regular graph of valency k and its second eigenvalue is related to expansion properties of the graph [1]. In this paper we investigate infinite graphs and show that in this case the expansion properties are related to the spectral radius of the graph. First we introduce necessary ...

Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

Periodic orbit theory and spectral statistics for scaling quantum graphs

The explicit solution to the spectral problem of quantum graphs found recently in [20], is used to produce the exact periodic orbit theory description for the probability distributions of spectral statistics, including the distribution for the nearest neighbor separations, s n = k n − k n−1 , and the distribution of the spectral oscillations around the average, δk n = k n − ¯ k n .