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Following a number of recent studies of resolvent and spectral convergence ofnon-uniformly elliptic families of differential operators describing thebehaviour of periodic composite media with high contrast, we study thecorresponding one-dimensional version that includes a "defect": an inclusion offixed size with a given set of material parameters. It is known that thespectrum of the purely periodic case without the defect and its limit, as theperiod $\varepsilon$ goes to zero, has a band-gap structure. We consider asequence of eigenvalues $\lambda_\varepsilon$ that are induced by the defectand converge to a point $\l_0$ located in a gap of the limit spectrum for theperiodic case. We show that the corresponding eigenfunctions are "extremely"localised to the defect, in the sense that the localisation exponent behaves as$\exp(-\nu/\varepsilon),$ $\nu>0,$ which has not been observed in the existingliterature. As a consequence, we argue that $\l_0$ is an eigenvalue of acertain limit operator defined on the defect only. In two- andthree-dimensional configurations, whose one-dimensional cross-sections aredescribed by the setting considered, this implies the existence of propagatingwaves that are localised to the defect. We also show that the unperturbedoperators are norm-resolvent close to a degenerate operator on the real axis,which is described explicitly.

Extreme localisation of eigenfunctions to one-dimensional high-contrast periodic problems with a defect