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

Cherdantsev, Mikhail; Cherednichenko, Kirill; Cooper, Shane

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.

Citation

Cherdantsev, M., Cherednichenko, K., & Cooper, S. (2018). Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect. SIAM Journal on Mathematical Analysis, 50(6), 5825-5856. https://doi.org/10.1137/17m112261x

• 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