内部具有不连续性的微分算子的谱分析

This project focuses on the spectral theory of differential operators with an infinite number of interior discontinuous points . To consider the problem we intends to construct a new Hilbert space framework by combining the transmission conditions, which are added to these interior discontinuous points. The emphasis of the project are deficiency index, description of self adjoint domains, dissipative extension, the necessary and sufficiency conditions on eigenvalues, Green functions, the relationship of discreteness of spectrum and coefficients of differential operators etc.. Combining the ergodicity of billiard flows, we will investigate the progressive natures (exponential behavior) of functions in the null space of differential operators with interior discontinuity. In the case, where the weight function changes sign, we will put the emphasis on the analysis of non-real eigenvalues of differential operators with interior discontinuous points. The oscillation of eigenfuction of differential operators with interior discontinuity，the relationship between the number of zero points of eigenfuctions and the transmission conditions added to interior discontinuous points, are the important parts of this project. By the view of symplectic spaces we investigate dissipative extension of this kind of differential operators, give the canonical forms of self-adjoint boundary conditions of differential operators in term of symplectic matrix, and then consider the influence of boundary parameters on the distribution of eigenvalues of high order differential operators (firstly we consider 4-order case).

本课题重点研究内部具有无穷多个不连续点微分算子的谱理论。拟结合在不连续点附加的转移条件，把问题放在一个新的Hilbert空间框架中来考虑，研究重点是这类算子的亏指数、自共轭域的刻画，耗散扩张，特征值的充要条件，Green函数，系数与谱的离散性的关系等。结合弹子流（billiard flows）的遍历性问题的研究，课题组拟研究一类与弹子动力系统相关的具有无穷多个转移条件的微分算子零空间函数的渐进性质(指数级收敛或发散）。对于权函数变号的具有内部不连续性微分算子的谱分析，重点放在其非实特征值分析。具有内部不连续性微分算子的特征函数的振动性，特征函数零点个数与在不连续点附加的转移条件的关系，也是本课题研究的一个重要内容。本课题还拟从辛空间的角度研究耗散扩张，应用辛矩阵研究高阶自共轭微分算子边界条件的标准型，在此基础上研究高阶微分算子（首先是四阶微分算子）边界条件参数变化对特征值分布的影响。

项目组主要围绕内部具有无穷多个不连续点的Sturm-Liouville（S-L）算子的自共轭扩张、亏指数、Friedrichs扩张以及谱的离散性分析；内部具有不连续性的不定S-L算子的谱分析；几类内部具有不连续性的微分算子耗散性及特征值关于问题的依赖性等多个方面开展研究工作. 项目还研究了利用辛几何方法给出最小算子的耗散和严格耗散扩张并且刻画其边界条件；周期S-L问题；微分算子自共轭边界条件的标准型；C-对称微分算子的自共轭域描述等微分算子领域的重要问题. 并且对于一类重要的PT-对称微分算子，从数学的角度给出其自共轭域描述。相关研究成果整理论文25篇在国内外期刊上发表或接收。其中，SCI索引论文15篇。