On the completeness of root vectors of a certain class of differential operators



The present paper is the first one in a series of two papers devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non-selfadjoint unbounded differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary-value problem (IBVP) for a non-homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with dissipative boundary condition at one end and with a heavy load at the other end; (c) IBVP for a three-dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping. In the second paper of the series the following problems will be considered: (a) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (b) IBVP for a coupled Euler–Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending-torsion vibration model); (c) IBVP for a system of two Timoshenko beams coupled through linear Van der Waals forces. The model of (c) describes vibrational motion of a double-walled carbon nanotube. In all of the above cases, the result has been obtained by using Krien’s Theorem on completeness of root vectors of a dissipative operator with a nuclear imaginary part. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim