2017/02/12 by Cherdantsev, Mikhail, Cherednichenko, Kirill, Cooper, Shane
#FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1702.03538
Following a number of recent studies of resolvent and spectral convergence of non-uniformly elliptic families of differential operators describing the behaviour 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 ł0 located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are "extremely" localised to the defect, in the sense that the localisation exponent behaves as exp(-ν/ε), ν>0, which has not been observed in the existing literature. As a consequence, we argue that ł0 is an eigenvalue of a certain limit operator defined on the defect only. 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 localised to 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.