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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 $\varepsilon$ 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.
Highlights
The question of whether a macroscopic perturbation of material properties in a periodic medium or structure induces the existence of a localized solution to the time-harmonic version of the equations of motion is of special importance from the physics, engineering, and mathematical points of view
We demonstrate that for eigenvalue sequences that converge to a point in \BbbR \setminu\bigcup \theta \sigma (A\theta ) the corresponding eigenfunctions are localized to a small neighborhood of the defect
Up to the leading order, the values of \lambda \varepsi are described by an eigenvalue of the weighted Neumann
Summary
The question of whether a macroscopic perturbation of material properties in a periodic medium or structure (periodic composite) induces the existence of a localized solution (bound state) to the time-harmonic version of the equations of motion is of special importance from the physics, engineering, and mathematical points of view. The present work is a study of localization properties for a class of composite media that has been the subject of increasing interest in the mathematics and physics literature recently, in view of its relation to the so-called metamaterials, e.g., manufactured composites possessing negative refraction properties It has been shown in [8] that the spectrum of a stratified high-contrast composite, represented mathematically by a one-dimensional periodic second-order differential equation, has an infinitely increasing number of gaps (lacunae) opening in the spectrum, in the limit of the small ratio \varepsi between the period and the overall size of the composite.
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