Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity

The completeness theorem of the eigenfunction systems for the product of two 2×2 symmetric operator matrices is proved. The result is applied to 4×4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.