Date of Award

Fall 2009

Project Type


Program or Major


Degree Name

Doctor of Philosophy

First Advisor

Marianna A Shobov


This dissertation is devoted to analytical study of a contemporary model of a double-walled carbon nano-tube. Carbon nano-tubes have been considered outstanding candidates to innovate and to promote emerging technologies, due to their remarkable chemical, mechanical, and physical properties. For these technologies, there is a need to develop mathematical models that capture the nature of the responses of these structures under a variety of physical conditions. Developing these models is challenging because the behavior lies on the borderline between classical and quantum systems. The main goal of the present dissertation is to prove mathematically rigorous results concerning the vibrational behavior (frequencies and mode shapes) of a double walled carbon nano-tube.

In the dissertation, we assume that the vibrational behavior of a double-walled carbon nano-tube can be modeled as a system of two nested Timoshenko beams connected through the distributed Van der Waals forces. The model is given as a coupled system of four partial differential equations of the hyperbolic type equipped with a four-parameter family of dynamical boundary conditions. We rewrite the system of four equations with the set of eight boundary conditions as the first order in time evolution equation in the state space of the system (which is a Hilbert space of Cauchy data). The dynamics generator of the evolution equation is an 8-by-8 matrix differential operator, whose spectrum is directly connected to the vibrational frequencies of the coupled nano-tubes.

It is shown that this dynamics generator is an unbounded non-selfadjoint operator, and is a relatively compact perturbation of a dynamics generator corresponding to a system of two uncoupled nano-tubes. The main non-selfadjoint operator has compact resolvent, and the set of the eigenvalues splits into distinct branches whose representation has been found. Explicit asymptotic formulas for four-branch discrete spectrum have been proved. Analytical results are in agreement with physical origin of the model and numerical data.