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

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.