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

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

2002-07-16

arxiv.org/abs/math-ph/0207020v1

Physics

Mathematical Physics

12 pages, 1 eps-figure, LaTeX

Scientific paper

We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space $L_2(\R^m)$, 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 Schr\"odinger operators, elliptic operators in divergence form, Laplace-Beltrami-operators, Schr\"odinger 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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