Asymptotical distribution of eigenvalues of dynamics generator governing vibrations of double-walled carbon nanotube model

Article type: Research Article

Authors: Shubov, Marianna A. | Rojas-Arenaza, Miriam

Affiliations: Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA. E-mails: [email protected], [email protected]

Note: [] Corresponding author.

Abstract: The paper is devoted to rigorous spectral analysis of recently developed mathematical model of a double-walled carbon nanotube. The model is governed by a system of four partial differential equations describing vibrations of two Timoshenko beams coupled through distributed van der Waals forces. The system is equipped with a four-parameter family of nonconservative boundary conditions. The corresponding initial boundary-value problem has been reduced to an evolution equation in the state space. The dynamics generator is a nonselfadjoint matrix differential operator with purely discrete spectrum. It is shown that the entire spectrum asymptotically splits up into four spectral branches. Asymptotical representation has been derived for the eigenvalues along each spectral branch as the number of an eigenvalue tends to infinity. To prove the results, a two-step procedure involving construction of the left and right reflection matrices has been used.

Keywords: carbon nanotubes, Timoshenko system, dynamics generator, matrix differential operator, reflection matrices, asymptotics of the spectrum

DOI: 10.3233/ASY-2010-0991

Journal: Asymptotic Analysis, vol. 68, no. 1-2, pp. 89-123, 2010