  We study an abstract family of asymptotically degenerating variational
problems. Those are natural generalisations of families of problems emerging
upon application of a rescaled Floquet-Bloch-Gelfand transform to resolvent
problems for high-contrast elliptic PDEs with highly oscillatory periodic
coefficients. An asymptotic analysis of these models leads us to a hierarchy of
approximation results with uniform operator-type error estimates under various
assumptions, satisfied by specific examples. We provide approximations for the
resolvents in terms of a certain `bivariate' operator which appears an abstract
generalisation of the two-scale limit operators for highly oscillatory
high-contrast PDEs. The resulting approximating self-adjoint operator,
providing tight operator error estimates, is the bivariate operator sandwiched
by a connecting operator which for a broad class of periodic problems
specialises to a new two-scale version of the classical Whittaker-Shannon
interpolation. An explicit description of the limit spectrum in the abstract
setting is provided, and new tight error estimates on the distance between the
original and limit spectra are established. Our generic approach allows us to
readily consider a wide class of asymptotically degenerating problems including
but also going beyond high-contrast highly oscillatory PDEs. The obtained
results are illustrated by various examples.

Analysis of PDEs;Spectral Theory