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

2002/07/16 by Rainer Hempel, Hempel, Rainer, Olaf Post +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.math-ph/0207020

12 pages, 1 eps-figure, LaTeX

arxiv created 2002/07/16 · arxiv updated 2009/11/30

Abstract

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 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.

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