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Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs

Kuchment, Peter and Kunyansky, Leonid A. (1999) Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs. Experimental Mathematics, 8 (1). pp. 1-28. ISSN 1058-6458.

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The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem - Δu= λ∈u, where the dielectric constant ∈(x) is a periodic function which assumes a large value ∈ near a periodic graph Σ 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.

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Additional Information:© 2012 A K Peters, Ltd. Work of both authors was partially supported by the NSF Grant DMS-961044 and by a DEPSCoR Grant administered through the ARO. The first author was also partially supported by an NSF EPSCoR Grant. The content of this article does not necessarily reflect the position or the policy of the federal government. The authors express their gratitude to Professors P. Exner, A. Figotin, A. Klein, and Dr. I. Ponomarev for helpful discussions. We are thankful to Professors V. Isakov, S. Molchanov, and Z. Sun for references to literature and to reviewers for important remarks.
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Army Research Office (ARO) DEPSCoR GrantUNSPECIFIED
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Mathematical Reviews number (MathSciNet)MR1685034
Zentralblatt MATH identifier0930.35112
Issue or Number:1
Classification Code:Primary Subjects: 78A60; Secondary Subjects: 35P05, 47F05, 78M25
Record Number:CaltechAUTHORS:20120111-075338043
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28743
Deposited By: Tony Diaz
Deposited On:11 Jan 2012 21:47
Last Modified:03 Oct 2019 03:35

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