  We study the asymptotic behaviour of eigenvalues and eigenfunctions of 2D
vibrating systems with mass density perturbed in a vicinity of closed curves.
The threshold case in which resonance frequencies of the membrane and thin
inclusion coincide or closely situated is investigated. The perturbed
eigenvalue problem can be realized as a family of self-adjoint operators acting
on varying Hilbert spaces. However the so-called limit operator which is
ultimately responsible for the asymptotics of eigenvalues and eigenfunctions is
non-self-adjoint and possesses the Jordan chains of length $2$. Apart from the
lack of self-adjointness, the operator has non-compact resolvent. As a
consequence, its spectrum has a complicated structure, for instance, the
spectrum contains a countable set of eigenvalues with infinite multiplicity.
