Membranes with thin and heavy inclusions: asymptotics of spectra

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.