历经二十多年的发展，辛弹性力学方法已经解决了许多实际力学问题，但是应用该方法研究力学问题时，很多情况下只能得到原力学方程的数值解而得不到其解析解。为了解决这类问题，李锐等学者提出了辛叠加方法，进而得到了一些各向同性矩形板问题的解析解，但目前未能应用辛叠加方法解决其它力学问题。为了研究推广辛叠加方法的应用范围，我们将辛叠加方法的理论基础归为一类特殊无穷维Hamilton算子辛本征函数系的完备性问题。在已有无穷维Hamilton算子谱理论的基础上，本项目将通过研究各种具体力学问题，并结合符号运算、算子矩阵及其扰动理论等知识，研究构建这一类特殊无穷维Hamilton算子辛本征函数系为完备的一般理论，为辛叠加方法提供理论依据，进而应用辛叠加方法得到更多力学实际问题的解析解。

After more than two decades of development, many practical mechanical problems has been solved by symplectic elasticity approach. However, in many cases, only the numerical solutions of the original mechanical equations can be obtained, and the analytical solution can not be obtained when applying the approach to study mechanics problems. In order to solve these problems, Li Rui et al. proposed the symplectic superposition method and obtained some analytical solutions of the isotropic rectangular plate problems. But the symplectic superposition method can not be used to solve other mechanical problems at present. In order to study and promote the application range of the symplectic superposition method, we regard the theoretical basis of symplectic superposition method as the completeness problems of symplectic eigenfunctions for a special class of infinite dimensional Hamiltonian operators. Based on the existing infinite dimensional Hamiltonian spectrum theory, we study various concrete mechanical problems by combining the knowledge of symbolic operation, operator matrix and perturbation theory in the project. It is a general theory that the symplectic eigenfunctions of this kind of special infinite dimensional Hamiltonian are constructed,which provides a theoretical basis for the symplectic superposition method. Furthermore, the analytical solutions of the more practical problems in mechanics will be given by symplectic superposition method.