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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. http://resolver.caltech.edu/CaltechAUTHORS:20120111-075338043

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Abstract

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.


Item Type:Article
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http://projecteuclid.org/euclid.em/1047477109PublisherUNSPECIFIED
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.
Funders:
Funding AgencyGrant Number
NSFDMS-961044
Army Research Office (ARO) DEPSCoR Grant UNSPECIFIED
NSF DEPSCoR Grant UNSPECIFIED
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Mathematical Reviews number (MathSciNet)MR1685034
Zentralblatt MATH identifier0930.35112
Classification Code:Primary Subjects: 78A60; Secondary Subjects: 35P05, 47F05, 78M25
Record Number:CaltechAUTHORS:20120111-075338043
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120111-075338043
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28743
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Jan 2012 21:47
Last Modified:11 Jan 2012 21:47

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