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### 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 ϵ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues λ^{ϵ} that are induced by the defect and converge to a point λ_{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(-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 language | English |
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Pages (from-to) | 5825-5856 |

Number of pages | 32 |

Journal | SIAM Journal on Mathematical Analysis (SIMA) |

Volume | 50 |

Issue number | 6 |

Early online date | 20 Nov 2018 |

DOIs | |

Publication status | Published - 31 Dec 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Extreme localization of eigenfunctions to one-dimensional high-contrast periodic problems with a defect'. Together they form a unique fingerprint.

## Projects

- 1 Finished

### Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory

Engineering and Physical Sciences Research Council

23/07/14 → 22/06/19

Project: Research council

## Profiles

## Cite this

*SIAM Journal on Mathematical Analysis (SIMA)*,

*50*(6), 5825-5856. https://doi.org/10.1137/17M112261X