Research On The Basis Properties Of Systems Of Generalized Eigenvectors Of Unbounded Hamiltonian Operators

In the early 1990 s,academician Wanxie Zhong put forward a symplectic systematic methodology to solve some bottleneck problems in solid mechanics.This method overcomes the difficulty of solving higher order governing PDEs and the subjective guess of the solution form by the traditional semi-inverse method,enlarges the range of analytical solution,and gets rapid development in many fields such as applied mechanics.The mathematical foundation of symplectic system approach depends on the block Schauder basis property of the system of generalized eigenvectors of the infinite dimensional Hamiltonian operators.Based on this property,some PDEs which have not been solved can be solved rationally.In this dissertation,we discuss the block Schauder basis properties of the system of generalized eigenvectors of some unbounded Hamiltonian operators from both theoretical and applied perspectives.The idea of theoretical research is to give the equivalent characterization of the block Schauder basis properties of the system of generalized eigenvectors of some abstract unbounded operator matrices,and then apply the theoretical results to the concrete mechanical models.The idea of application research is to transform some specific mechanical equations into the equivalent infinite dimensional Hamiltonian systems,and prove that the system of generalized eigenvectors of corresponding Hamiltonian operators to be a block Schauder basis of some Hilbert space,and then give the analytical solutions of the original problems.In terms of theoretical research,firstly,we consider the block Schauder basis property of the system of generalized eigenvectors of a class of 2 × 2 Hamiltonian operator matrices.An equivalent relation is established between the block Schauder basis property of the system of generalized eigenvectors of this kind of operator matrices and the Schauder basis property of the system of generalized eigenvectors of quadratic operator pencil induced from the former.Furthermore,it is shown that the system of generalized eigenvectors of a class of 4 × 4 Hamiltonian operators derived from the bending problem of rectangular thin plate with two opposite sides simply supported is a block Schauder basis of corresponding Hilbert space.Secondly,we discuss the block Schauder basis property of the system of generalized eigenvectors of a class of 3 × 3 operator matrices.An equivalent description is obtained between the block Schauder basis property of the system of generalized eigenvectors of this kind of operator matrices and the Schauder basis properties of the system of generalized eigenvectors of two kinds of operator pencils induced from this kind of operator matrices.As applications,the block Schauder basis properties of the system of generalized eigenvectors of the product operators of off-diagonal blocks of two classes of 6×6 off-diagonal Hamiltonian operators derived from the rectangular moderately thick plate problems with two opposite sides simply supported are investigated.Then,we study the block Schauder basis property of the system of generalized eigenvectors of a class of 4 × 4 Hamiltonian operator matrices.A necessary and sufficient condition is given for the system of generalized eigenvectors of this kind of operator matrices to be a block Schauder basis of some Hilbert space,and the obtained results are applied to the free vibration and bending problems of rectangular thin plates with two opposite sides simply supported.In terms of applied research,first of all,we use the symplectic system approach to establish a unified framework for solving a class of PDEs arising from the elasticity,which focus on applications to plate structures.By introducing appropriate state functions,this kind of PDEs are transformed into an equivalent separable infinite dimensional Hamiltonian system.It is proved that the system of generalized eigenvectors of corresponding off-diagonal Hamiltonian operator is a block Schauder basis of some Hilbert space,which provides a theoretical guarantee for the successful implementation of symplectic system approach.The analytical solutions for bending,buckling and free vibration problems of fully clamped rectangular thin plates governed by this kind of PDEs are obtained by utilizing the theorem of basis property and the technique of symplectic superposition.Numerical examples are presented to verify the correctness of the analytical solutions.It is worth mentioning that the plane elasticity problems of two-dimensional octagonal quasicrystals are analyzed by using symplectic system approach.Within the framework of the Hamiltonian system in the symplectic space,we obtain the analytical solutions to the plane elasticity problem of point group 8mm octagonal quasicrystals.Subsequently,the comparison studies of numerical results are conducted to confirm the convergence and accuracy of the analytical solutions.In addition,we derive the infinite dimensional Hamiltonian system and final governing equation for the challenging plane elasticity problem of Laue class 15 of octagonal quasicrystals,which would be useful for further solving the problem with the help of symplectic system approach or semi-inverse method.The methods presented in this dissertation can be employed as references for the study of some applied mechanical models and the solution of some PDEs.The relevant conclusions offer a theoretical guarantee for the method of separation of variables under the framework of Hamiltonian system.Some newly obtained analytical solutions can be served as benchmarks for verifying other numerical methods.