Membranes with thin and heavy inclusions: Asymptotics of spectra

Article type: Research Article

Authors: Golovaty, Yuriy^{; *}

Affiliations: Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, Universytetska str.1, Lviv, 79000, Ukraine. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide 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 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. The complete asymptotic analysis of eigenvalues has been carried out.

Keywords: Asymptotics of eigenvalues, eigenvalue of infinite multiplicity, quasimode, non-self-adjoint operator, concentrated mass, singular perturbation

DOI: 10.3233/ASY-211743

Journal: Asymptotic Analysis, vol. 130, no. 1-2, pp. 23-51, 2022