无界斜对角Hamilton算子及其在辛弹性力学中的应用

Infinite-dimensional separable Hamiltonian systems have important applications in various aspects of symplectic elasticity such as plane elasticity, plate bending problems, orthotropic anisotropic materials and so on. In these applications, the off-diagonal Hamiltonian operators and the product operator of two symmetric operator matrices play a central role. The motivation of this project is to obtain better spectral properties of these operators via decoupling some symplectic elasticity problems. To achieve the goal, we need to find the close relationship of the spectral properties between the off-diagonal Hamiltonian operators and the product operator of two symmetric operator matrices, to disclose the essential difference between biorthogonality and symplectic orthogonality in the orthogonal normalized process, and to analyze the block Schauder basis properties of generalized eigenfunction system of the product operator of two symmetric operator matrices by Fourier analysis and Titchmarsh methods, so as to get the symplectic analytic solutions for the studied problems via biorthogonal expansion method. Furthermore, we need to establish some criteria for closedness of range, invertibility, and generalized invertibility of the symmetric operator matrices to serve the needs of decoupling separable Hamiltonian system and biorthogonal expansion method. The success of this project will enrich the theory of symplectic elasticity and build a secure mathematical foundation for the relevant practical problems in mechanics.

无穷维可分Hamilton 系统在平面弹性、板弯曲问题、正交各向异性材料等辛弹性力学领域有重要应用，在各种应用中无界斜对角Hamilton算子和对称算子矩阵的乘积起着决定作用。本项目拟从算子理论角度深入研究无穷维可分Hamilton 系统，将某些辛弹性力学问题进行解耦并研究对称算子矩阵乘积的谱理论。具体地，建立斜对角Hamilton算子和对称算子矩阵乘积的谱理论之间的完全联系；探讨双正交和辛正交关系在归一性方面的重要区别；运用Fourier 分析和Titchmarsh 方法研究对称算子矩阵乘积的广义本征函数系的块状基性质，通过双正交展开方法最终获得辛弹性力学问题的辛解析解；运用空间分解技巧和分块算子理论研究对称算子矩阵的值域闭性、可逆性和广义可逆性，为可分Hamilton系统的解耦和双正交展开解法服务。本项目的成功实施不仅能丰富辛弹性力学的研究内容，还能为力学等领域的相关实践提供数学基础。

本项目从算子理论角度出发，深入研究了无穷维可分 Hamilton 系统。结合对称算子乘积的性质，得到了无界斜对角Hamilton算子本征函数系间的一种双正交关系，讨论了双正交和辛正交之间的联系与区别，进而运用Fourier分析方法证明了无界斜对角Hamilton算子广义本征函数系的块状Schauder基性质；根据对称算子矩阵乘积后的形式，研究了对称算子矩阵乘积的本征函数系的完备性，并应用于对边简支的辛弹性力学问题。建立了矩形中厚板弯曲问题的双正交展开解法，并给出了此问题的辛解析解。本项目运用Riesz积分方法探讨了Banach代数上幂等算子线性组合的广义可逆性，推广了前人的相关工作；通过分析Hamilton算子的值域与零空间、定义域的形式，研究了一般的Hamilton算子的可逆性，给出了Hamilton算子可逆性研究的一般方法和统一框架。项目组通过点谱和剩余谱的相互刻画技术，结合空间分解技巧等，给出了有界线性算子和其Aluthge变换及3×3阶算子矩阵的谱分析。利用算子矩阵的Frobenius-Schur分解，刻画了Hamilton算子矩阵的辛自共轭性质，并应用于辛弹性力学问题，还从2×2分块算子矩阵角度出发，对Hamilton算子矩阵的辛自共轭性进行了分类等。本项目的研究成果一定程度上为钟万勰院士的专著《弹性力学求解新体系》提供了理论基础。项目执行期间已公开发表学术论文17篇，其中SCI检索的杂志上有5篇。