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

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 $exp(-nu/varepsilon),$ $nuu003e0,$ which has not been observed in the existing literature. In two- and three-dimensional configurations, whose one-dimensional cross sections are...