无界Hamilton算子谱的渐近性与广义本征函数系的块状基性质研究

弹性力学辛体系方法的应用已取得重要进展，但其数学理论的研究还不够完善。本项目拟深入研究无界Hamilton算子的谱分布和点谱的渐近性，从多个角度刻画其广义本征函数系的块状基性质：利用Hamilton算子矩阵内部算子的特性，研究其谱的分布、预解集非空、Hilbert-Schmidt性、紧性、点谱和广义本征函数系的渐近性；研究多项式算子族的谱性质与广义本征函数系的基性质，刻画Hamilton算子广义本征函数系的块状基性质；从积分方程角度，建立Hamilton核积分算子的本征值理论；利用有限差分理论，研究离散Hamilton算子的谱性质与广义本征函数系的块状基性质，逼近无界Hamilton算子的相应性质；综合运用算子理论、有限差分和复分析方法，对辛体系中有代表性的无界Hamilton算子进行相应研究。本项目的成功实施，不仅能为辛体系方法提供理论基础，而且能为某些非自伴问题的研究提供新的思路。

Applications of symplectic elasticity method have been made important progress, but the study of mathematical theory of the method is far from perfect. This project will investigate the spectral descriptions and asymptotics of the point spectrums, and then describe the block basis properties of generalized eigenfunction systems of unbounded Hamiltonian operators (UHOs) by differential ways comprehensive in this project. This project will focus on the following topics: using the characteristics of the entries of Hamiltonian operator matrices, study the spectral descriptions, the resolvent set to be nonempty, Hilbert-Schmidt property, compactness, asymptotics of the point spectrums and generalized eigenfunction systems of UHOs; consider the spectral properties and the block basis properties of generalized eigenfunction systems of UHOs via the analysis of the relevant properties of the polynomial operator pencils; establish the eigenvalue theory of integral operators with the Hamiltonian kernel from the perspective of integral equations; applying the finite difference theory, investigate the spectral properties and block basis properties of the generalized eigenfunction systems of the discrete Hamiltonian operators, and then approach the corresponding properties of UHOs; study the related properties of the typical UHOs arising from the symplectic elasticity models by using operator theory, finite difference and complex analysis method. The successful implementation of this project can not only provide theoretical foundation for symplectic elasticity but also offer some references for the study of some non-self-adjoint problems.