Eigen-Vector Expansion Theorem Of A Class Of Operator Matrices Arising In Elasticity And Applications

The eigen-vector expansion theorem is one of the most important conclusions in the spectral theory of linear operators, and it is the theoretical foundation of the variables separation method in solving mathematical physics problem. The classical variables sep-aration method requires the symmetry of the considered problems, i.e., their fundamental equations and related boundary value conditions are all self-adjoint. This greatly limit the application areas of the method. In this paper, we begin with a summary of the back-ground and recent advances of non-self-adjoint operators. Next, we consider the properties of the expansions in terms of eigen-vectors for a class of non-self-adjoint operators arising in elasticity; it is shown that the eigen-vectors system possesses the symplectic orthogo-nality, and a necessary and sufficient condition for the completeness of the eigen-vector system is further given. Finally, the obtained results are applied to the concrete problems, and the effectiveness and correctness of the results are demonstrated.