  A model of strongly inhomogeneous medium with simultaneous perturbation of
rigidity and mass density is studied. The medium has strongly contrasting
physical characteristics in two parts with the ratio of rigidities being
proportional to a small parameter $\epsilon$. Additionally, the ratio of mass
densities is of order $\epsilon^{-1}$. We investigate the asymptotic behaviour
of spectrum and eigensubspaces as $\epsilon\to 0$. Complete asymptotic
expansions of eigenvalues and eigenfunctions are constructed and justified.
  We show that the limit operator is nonself-adjoint in general and possesses
two-dimensional Jordan cells in spite of the singular perturbed problem is
associated with a self-adjoint operator in appropriated Hilbert space
$L_\epsilon$. This may happen if the metric in which the problem is
self-adjoint depends on small parameter $\epsilon$ in a singular way. In
particular, it leads to a loss of completeness for the eigenfunction
collection. We describe how root spaces of the limit operator approximate
eigenspaces of the perturbed operator.
