Spectral Gaps for Periodic Elliptic Operators with High Contrast: an Overview

On Spectrum Gaps of Some Divergent Elliptic Operators with Periodic Coefficients

More than 10 years ago physicists gave a theoretic description of the so-called photonic crystal, an optic analog of a semiconductor. In contrast to a semiconductor, the photonic crystal is an artificial material, a composite. The dominant requirement for the photonic crystal is that electromagnetic waves of a certain length cannot propagate in it. It was also predicted that the photonic crysta...

On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators

Asymptotic distribution of eigenvalues of the elliptic operator system

Since the theory of spectral properties of non-self-accession differential operators on Sobolev spaces is an important field in mathematics, therefore, different techniques are used to study them. In this paper, two types of non-self-accession differential operators on Sobolev spaces are considered and their spectral properties are investigated with two different and new techniques.

An Overview of Periodic Elliptic Operators

The article surveys the main topics, techniques, and results of the theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which significantly influence most spectral features of such operators. The approaches described are applicable not only to the standard model example of Schrödinger...

The spectral properties of differential operators with matrix coefficients on elliptic systems with boundary conditions

Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...