相关论文: Degeneracies in the length spectra of metric graph…

Many quantum systems may have the same classical limit. We argue that in the classical limit their traces do not necessarily converge one to another. The trace formula allows to express quantum traces by means of classical quantities as…

We study the spectral statistics of quantum (metric) graphs whose vertices are equipped with preferred orientation vertex conditions. When comparing their spectral statistics to those predicted by suitable random matrix theory ensembles,…

We make connections of a counting problem of Eulerian cycles for undirected graphs to homological spectral graph theory, and formulate explicitly a trace formula that identifies the number of Eulerian circuits on an Eulerian graph with the…

We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically…

We analyze spectral properties of a quantum graph in the form of a ring chain with a $\delta$ coupling in the vertices exposed to a homogeneous magnetic field perpendicular to the graph plane. We find the band spectrum in the case when the…

We consider quantum walks on a finite graphs to which infinite tails are attached. We explore how the propagating and bound states depend on the structure of the finite graph. The S-matrix for such graphs is defined. Its unitarity is proved…

During the last years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wave function statistics. In the first part of this review we give a detailed introduction to…

We discuss spectral properties of an periodic quantum graph consisting of an array of rings coupled either tightly or loosely through connecting links, assuming that the vertex coupling is manifestly non-invariant with respect to the time…

Complex networks or graphs are ubiquitous in sciences and engineering: biological networks, brain networks, transportation networks, social networks, and the World Wide Web, to name a few. Spectral graph theory provides a set of useful…

We connect quantum graphs with infinite leads, and turn them to scattering systems. We show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics: typical poles, delay…

We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wavefunction statistics. This operator is the analogue of the classical evolution operator on…

Following the derivation of the trace formulae in the first paper in this series, we establish here a connection between the spectral statistics of random regular graphs and the predictions of Random Matrix Theory (RMT). This follows from…

A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive…

We prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations, as predicted by random--matrix theory. The stability of these correlations with regard to non--universal corrections is analyzed in…

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case…

Given an unbalanced open quantum graph, we derive a formula relating sums over its scattering resonances with integrals outside a strip. We deduce lower bounds on the number of resonances (in bounded regions of the complex plane),that are…

We introduce a simple diagrammatic approach for estimating how a randomly walking quantum particle searches on a graph in continuous-time, which involves sketching small weighted graphs with self-loops and considering degenerate…

One can define the notion of primitive length spectrum for a simple regular periodic graph via counting the orbits of closed reduced primitive cycles under an action of a discrete group of automorphisms. We prove that this primitive length…

This work explores the spectra of quantum graphs where the Schr\"odinger operator on the edges is equipped with a potential. The scattering approach, which was originally introduced for the potential free case, is extended to this case and…

The unitary evolution maps in closed chaotic quantum graphs are known to have universal spectral correlations, as predicted by random matrix theory. In chaotic graphs with absorption the quantum maps become non-unitary. We show that their…