Membranes with thin and heavy inclusions: Asymptotics of spectra

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