Experimental Mathematics

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

Peter Kuchment and Leonid A. Kunyansky


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.

Article information

Experiment. Math. Volume 8, Issue 1 (1999), 1-28.

First available in Project Euclid: 12 March 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 78A60: Lasers, masers, optical bistability, nonlinear optics [See also 81V80]
Secondary: 35P05: General topics in linear spectral theory 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47) 78M25: Other numerical methods


Kuchment, Peter; Kunyansky, Leonid A. Spectral properties of high contrast band-gap materials and operators on graphs. Experiment. Math. 8 (1999), no. 1, 1--28. http://projecteuclid.org/euclid.em/1047477109.

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