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Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect

Cherdantsev, Mikhail; Cherednichenko, Kirill; Cooper, Shane

Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect Thumbnail


Authors

Mikhail Cherdantsev

Kirill Cherednichenko

Shane Cooper



Abstract

Following a number of recent studies of resolvent and spectral convergence of nonuniformly elliptic families of differential operators describing the behavior of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a “defect”: an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $\lambda_\varepsilon$ that are induced by the defect and converge to a point $\lambda_0$ located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are “extremely” localized to the defect, in the sense that the localization exponent behaves as $\exp(-\nu/\varepsilon),$ $\nu>0,$ which has not been observed in the existing literature. In two- and three-dimensional configurations, whose one-dimensional cross sections are described by the setting considered, this implies the existence of propagating waves that are localized to a vicinity of the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly.

Journal Article Type Article
Acceptance Date Aug 22, 2018
Online Publication Date Nov 20, 2018
Publication Date Nov 20, 2018
Deposit Date Jan 4, 2019
Publicly Available Date Jan 4, 2019
Journal SIAM Journal on Mathematical Analysis
Print ISSN 0036-1410
Electronic ISSN 1095-7154
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 50
Issue 6
Pages 5825-5856
DOI https://doi.org/10.1137/17m112261x
Public URL https://durham-repository.worktribe.com/output/1306037

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Copyright Statement
© 2018, Society for Industrial and Applied Mathematics.





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