Extreme localization of eigenfunctions to one-dimensional high-contrast periodic problems with a defect

Mikhail Cherdantsev, Kirill Cherednichenko, Shane Cooper

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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 ϵ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues λϵ that are induced by the defect and converge to a point λ0 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(-ν/ϵ), ν > 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.

Original languageEnglish
Pages (from-to)5825-5856
Number of pages32
JournalSIAM Journal on Mathematical Analysis (SIMA)
Issue number6
Early online date20 Nov 2018
Publication statusPublished - 31 Dec 2018


  • Decay estimates
  • High-contrast homogenization
  • Spectrum
  • Wave localization

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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