Related papers: Spectral correspondences for finite graphs without…
Let $(X,E_X)$ and $(V,E_V)$ be finite connected graphs without loops. We assume that $V$ has two distinguished vertices $a,b$ and an automorphism $\gamma$ which exchanges $a$ and~$b$. The $V$-edge substitution of $X$ is the graph $X[V]$…
The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…
We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite approximation'' method. These spectra are simple…
We introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries. The method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way…
We introduce a transfer matrix method for the spectral analysis of discrete Hermitian operators with locally finite hopping. Such operators can be associated with a locally finite graph structure and the method works in principle on any…
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…
Spectral properties of many finite convolution integral operators have been understood by finding differential operators that commute with them. In this paper we compile a complete list of such commuting pairs, extending previous work to…
On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity…
In this paper we introduce a spectra preserving relation between graphs with loops and graphs without loops. This relation is achieved in two steps. First, by generalizing spectra results got on (m, k)-stars to a wider class of graphs, the…
We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schr\"odinger operator on a star-graph with a finite number of…
We develop the method of similar operators to study the spectral properties of unbounded perturbed linear operators that can be represented by matrices of various kinds. The class of operators under consideration includes various…
The paper deals with first order self-adjoint elliptic differential operators on a smooth compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on…
We consider periodic Schr\"{o}dinger operators on the hexagonal lattice with self-adjoint vertex conditions that allow discontinuity and concentrated mass at the vertices. This model generalizes the periodic Schr\"{o}dinger operator on the…
We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the `energy'--average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric non--linear $\sigma$--model…
We describe some classes of linear operators on Banach spaces over non-Archimedean fields, which admit orthogonal spectral decompositions. Several examples are given.
Periodic $2$nd order ordinary differential operators on $\R$ are known to have the edges of their spectra to occur only at the spectra of periodic and antiperiodic boundary value problems. The multi-dimensional analog of this property is…
Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at…
In a finite-dimensional Euclidian space we consider a connected metric graph with the following property: each two cycles can have at most one common point. Such graphs are called A-graphs. On noncompact A-graph we consider a scattering…
We discuss ways in which momentum operators can be introduced on an oriented metric graph. A necessary condition appears to the balanced property, or a matching between the numbers of incoming and outgoing edges; we show that a graph…
The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between finite-dimensional and infinite-dimensional spaces. We first…