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Spectral properties of high contrast band-gap materials and operators on graphs

Peter Kuchment and Leonid A. Kunyansky
Source: Experiment. Math. Volume 8, Issue 1 (1999), 1-28.

Abstract

The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem

$$%-\Delta u=\lambda\varepsilon u$$%,

where the dielectric constant $\varepsilon(x)$ is a periodic function which assumes a large value $\varepsilon$ near a periodic graph $\Sigma$ in $\R^2$ and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of the spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.

First Page:
Primary Subjects: 78A60
Secondary Subjects: 35P05, 47F05, 78M25
Full-text: Open access
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.em/1047477109
Mathematical Reviews number (MathSciNet): MR1685034
Zentralblatt MATH identifier: 0930.35112

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