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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 onedimensional 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 bandgap 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 threedimensional configurations, whose onedimensional 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 normresolvent close to a degenerate operator on the real axis, which is described explicitly.
Original language  English 

Pages (fromto)  58255856 
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
 Highcontrast homogenization
 Spectrum
 Wave localization
ASJC Scopus subject areas
 Analysis
 Computational Mathematics
 Applied Mathematics
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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