  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 $\varepsilon$ 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 $\l_0$ 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
$\exp(-\nu/\varepsilon),$ $\nu>0,$ which has not been observed in the existing
literature. As a consequence, we argue that $\l_0$ 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.
