辛本征函数系的完备性及其在力学中的应用

Is it appropriate and correct that the method of separation of variables based on Hamiltonian system? This is the key problem of symplectic elasticity approach, and the theoretical basis of the above problem can come down to the completeness of symplectic eigenfunction systems (the eigenfunction systems of infinite dimensional Hamiltonian operators). But up to now, the completeness of symplectic eigenfunction systems is partly solved and a relatively perfect theory system for the completeness has not been set up. In this project, by applying complex analysis, symbolic computation, functional analysis and operator theory, we will study the completeness of the eigenfunction systems of various infinite dimensional Hamiltonian operators which are derived by various types of partial differential equations in mechanics. Then we will try to find the rule for the completeness of symplectic eigenfunction systems. Combining with operator matrix and its perturbation theory, we will try to establish the completeness theory of symplectic eigenfunction systems, which can be described by spectral properties of infinite dimensional Hamiltonian operator. It can provide theoretical basis for the symplectic eigenfunction expansion method, i.e., the variable separation methods based on the Hamiltonian system. Furthermore, based on the obtained theory, we will try to solve more practical problems in mechanics applying symplectic elasticity approach.

辛弹性力学方法所面临的关键问题是在Hamilton体系下采用分离变量法是否合适和正确？而这个问题的理论基础可归为辛本征函数系（无穷维Hamilton算子的本征函数系）的完备性问题。目前，辛本征函数系的完备性问题只得到部分解决，还没有形成一套比较完善的理论体系。本项目结合复分析、符号运算、泛函分析和算子理论等，研究由力学中的各种偏微分方程（组）所导出的各种具体无穷维Hamilton算子本征函数系的完备性，寻找辛本征函数系的完备性规律。再结合算子矩阵及其扰动理论等，研究构建用无穷维Hamilton算子谱的性质刻画的辛本征函数系的完备性理论。为辛本征函数展开法，即Hamilton体系下的分离变量法提供理论基础，进而在所得理论的指引下，应用辛弹性力学方法解决力学中的更多实际问题。

上世纪90年代，钟万勰院士将 Hamilton 体系引入到弹性力学，建立了辛弹性力学方法。目前辛弹性力学方法在弹性力学、流体力学、功能梯度材料、热效果、控制理论、波的传播及振动等诸多领域中均有广泛的应用。辛弹性力学方法的理论基础可归为无穷维 Hamilton 算子的谱理论问题，特别是无穷维 Hamilton 算子辛本征函数系的完备性问题。本项目按照研究计划结合符号运算、泛函分析和算子理论等知识，以无穷维 Hamilton 算子辛本征函数系的完备性为主线，展开了系列的研究工作。研究了各向同性、正交各向异性、弹性地基上正交各向异性及双参数弹性地基上正交各向异性矩形薄板弯曲（和受迫振动）问题等所对应的各种具体无穷维 Hamilton 算子，并分别证明这些无穷维 Hamilton 算子辛本征函数系在 Cauchy 主值意义下的完备性；研究了无穷维 Hamilton 算子的近似点谱、对角定义的上三角无穷维 Hamilton 算子谱的性质和某类无穷维 Hamilton 算子的 Moore-Penrose 广义逆等内容；在此基础上，探索了无穷维 Hamilton 算子与辛本征函数系完备性相关的谱理论。所得一些研究结果，如，正交各向异性矩形薄板弯曲（和受迫振动）问题及双参数弹性地基上正交各向异性矩形薄板弯曲（和受迫振动）问题所对应无穷维 Hamilton 算子辛本征函数的完备性等结果，拓宽了辛弹性力学方法的应用范围。本项目研究结果在一定程度上丰富了无穷维 Hamilton 算子本征函数系完备性的理论，为力学和应用力学实际问题提供了理论依据。