Related papers: Relationship between scattering matrix and spectru…
Recently, QCD Dirac spectra have been obtained for reasonably large lattices. We argue that correlations of these spectra are universal and can be obtained from a random matrix model with the global symmetries of QCD. Analytical arguments…
We consider the Schroedinger operator on graphs and study the spectral statistics of a unitary operator which represents the quantum evolution, or a quantum map on the graph. This operator is the quantum analogue of the classical evolution…
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 Laplace-Beltrami operator in the curved M\"obius strip is investigated in the limit when the width of the strip tends to zero. By establishing a norm-resolvent convergence, it is shown that spectral properties of the operator are…
In this paper we investigate the spectral and scattering theory for operators acting on topological crystals and on their perturbations. A special attention is paid to perturbations obtained by the addition of an infinite number of edges,…
The paper is devoted to investigation of the spectrum of a perturbed Laplace-Beltrami operator on manifolds with closed geodesics of the same length and where metric is a perturbation of the metric of the unit sphere. As the result, we give…
There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to…
We prove a fixed frequency inverse scattering result for the magnetic Schr\"odinger operator (or connection Laplacian) on surfaces with Euclidean ends. We show that, under suitable decaying conditions, the scattering matrix for the operator…
The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf…
Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…
We consider the Schroedinger operator with a complex delta interaction supported by two parallel hypersurfaces in the Euclidean space of any dimension. We analyse spectral properties of the system in the limit when the distance between the…
We consider the Laplace operator in a planar waveguide, i.e., an infinite two-dimensional straight strip of constant width, with particular types of Robin boundary conditions. We study the essential spectrum of the corresponding Laplacian…
For a large class of quantum systems the statistical properties of their spectrum show remarkable agreement with random matrix predictions. Recent advances show that the scope of random matrix theory is much wider. In this work, we show…
We consider the density matrices derived from combinatorial laplacian matrix of graphs. Specifically, the star-relevant graph, which means adding certain edges on peripheral vertices of star graph, is the focus of this paper. Initially, we…
We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the…
We connect quantum compact graphs with infinite leads, and turn them into scattering systems. We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph. The…
Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. It is proposed here to generalise this approach by considering arbitrary, directed graphs…
We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of…
We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-)symmetric form, thus characterizing (skew-)self-adjoint and unitary operators by means of self-ortho-gonal subspaces. By orthogonality…
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and…