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

Abstract

We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space $L_2({\mathbb{R}}^m)$, for $m\ge 2$. 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 operators, elliptic operators in divergence form, Laplace-Beltrami-operators, Schrödinger and Pauli operators with periodic magnetic fields. There are corresponding applications in heat and wave propagation, quantum mechanics, and photonic crystals.

12 pages, 1 eps-figure, LaTeX

Cited by in corpus (6)

• Spectral gaps for periodic Schrödinger operators with strong magnetic fields
• Honeycomb Schroedinger operators in the strong binding regime
• Existence of spectral gaps, covering manifolds and residually finite groups
• Example of periodic Neumann waveguide with gap in spectrum
• Gap opening and split band edges in waveguides coupled by a periodic system of small windows
• On eigenvalue and eigenvector estimates for nonnegative definite operators