On Spectrum Gaps of Some Divergent Elliptic Operators with Periodic Coefficients

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...

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

We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space L2(R m), for m ≥ 2. We specifically consider situations where high contrast in the coefficients leads to weak coupling between the period cells. Weak coupling of periodic systems frequently produces spectral gaps or spectral concentration. Our examples include Schrödinger ...

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

On the Bethe-sommerfeld Conjecture for the Polyharmonic Operator

It is a general property of elliptic differential operators with periodic coefficients, that their spectra are formed by union of closed intervals called spectral bands (see [12], [14]) possibly separated by gaps. One of the challenging questions of the spectral theory of periodic operators is to find out whether or not the number of gaps in the spectrum of a given operator is finite. The state...

Spectral convergence for high contrast elliptic periodic problems with a defect via homogenisation

We consider an eigenvalue problem for a divergence form elliptic operator Aε with high contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of ‘order one’ size. The local perturbation of coefficients for such operator could result in emergence of localised waves eigenfunctions with corresponding eigenva...