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We study the spectra for a class of differential operators with asymptotically constant coefficients.These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We…
As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell…
This paper is devoted to the description of our recent results on the spectral behavior of one-dimensional adiabatic quasi-periodic Schrodinger operators. The specific operator we study is a slow periodic perturbation of an incommensurate…
We describe the spectral properties of the Jacobi operator $(Hy)_n= a_{n-1} y_{n-1}+a_{n}y_{n+1}+b_ny_n,$ $n\in\Z,$ with $a_n=a_n^0+ u_n,$ $b_n= b_n^0+ v_n,$ where sequences $a_n^0>0,$ $b_n^0\in\R$ are periodic with period $q$, and…
We present a tight-binding investigation of strained bilayer graphene within linear elasticity theory, focusing on the different environments experienced by the A and B carbon atoms of the different sublattices. We find that the…
We consider the one-dimensional discrete Schr\"odinger operator with complex-valued sparse periodic potential. The spectrum for a complex-valued periodic potential is a complicated compact set in the complex plane represented by real…
We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of "slowing down" the twisting in the mean gives rise to a non-empty…
We consider the third order operator with small 1-periodic coefficients on the real line. The spectrum of the operator is absolutely continuous and covers all real line. Under the minimal conditions on the coefficients we show that there…
Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form $T_0 + i \gamma s_n$ on a Hilbert space, where $s_n$ is strongly convergent to the identity operator and $\gamma > 0$. We work in…
The discrete spectrum of complex banded matrices which are compact perturbations of the standard banded matrix of order $p$ is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides…
We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Watts-Strogatz small-world model, we employ a single parameter to determine the probability of local connections…
Let $(\cX, \rho)$ be a discrete metric space. We suppose that the group $\sZ^n$ acts freely on $X$ and that the number of orbits of $X$ with respect to this action is finite. Then we call $X$ a $\sZ^n$-periodic discrete metric space. We…
The spectral problem on a periodic domain with cracks is studied. An asymptotic form of dispersion relations is calculated under assumption that the opening of the cracks is small.
We study Schr\"odinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by $\delta$-couplings with a parameter $\alpha\in\R$. If the graph is "straight", i.e. periodic…
The spectrum of triangular band matrices defined on the sequence spaces where the entries of each band is a constant or convergent sequence is well studied. In this article, the spectrum and fine spectrum of a new generalised difference…
In this paper we discuss the spectral properties of one-term symmetric differential operators of even order with interior singularity, namely, we determine the deficiency numbers, describe its self-adjoint extensions and their spectrum. It…
In this paper, under some integrability condition, we prove that an electrical perturbation of the discrete Dirac operator has purely absolutely continuous spectrum for the one dimensional case. We reduce the problem to a non-self-adjoint…
We study the effects of adding a local perturbation in a pattern forming system, taking as an example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a…
The present work discusses the impact of vacancy defects in solid high-order harmonic generation. The total energy cutoff of the high-order harmonic spectrum increases as a function of concentration of vacancy defects, and the total…
We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks…