微分方程的解对微分算子自共轭域和谱的离散性的影响

The spectral analysis of the differential operators, especially of the discreteness of spectrum of differential operators , has significant meaning in both theory field and application field. In this project we will investigate the influence of solutions of differential equations on self-adjoint domains and discreteness of spectrum of differential operators. For the ordinary differential operators with complex coefficients, we will give the classification of the self-adjoint boundary conditions, characterize the self-adjoint domains and discuss the influence of the number and property of real-parameter square-integrable solutions in some open interval on the discreteness of spectrum of differential operators. And we will we will give the description of the self-adjoint domains and discuss the distribution of its spectrum by using the latest methods from symplectic geometry and Limit-Circle solutions. In this project, we will systematically investigate the operator with transmission conditions, and will investigate the distribution of spectrum of Schrödinger operators in terms of the theory of the singular differential operators with transmission conditions and Weyl function. Another important aim of our project is to give the canonical forms of general ordinary differential operators of any order, even or odd, with arbitrary deficiency index. Because of the connection between the self-adjoint boundary conditions and the solutions of differential operators, we will discuss the dependence of eigenvalues of the general ordinary differential operators of any order, firstly of the order four, on the self-adjoint boundary conditions.

微分算子的谱分析特别是谱的离散性研究，无论从理论上还是应用上都有十分重要的意义。本课题拟围绕微分方程的解对微分算子自共轭域以及谱的离散性的影响展开研究。对于复系数微分算子，拟重点研究奇数阶实参数平方可积解的分类和由其刻画的自共轭域，进而研究和探讨在某个区间上实参数平方可积解的个数和性质对微分算子谱分布的影响。并且我们拟从辛几何的角度结合微分方程解对其定义域的刻画研究对称算子的耗散扩张及其谱的分布。课题组计划深入研究内部有不连续点的算子，并利用带有转移条件的奇异微分算子的自共轭理论和Weyl 函数理论研究势函数是广义函数的薛定谔算子的谱分布。课题组研究的另外一个重要目标是给出任意阶实系数或复系数微分算子自共轭域的标准型，由于这样的边界条件是和微分方程的解相联系，进而可以从边界条件的标准型入手研究任意阶自共轭微分算子，首先是四阶微分算子边界条件中参数变化对特征值分布的影响。

本课题主要围绕实参数解刻画的微分算子自伴域以及微分算子谱的离散性展开研究。项目组使用微分方程实参数解刻画了奇数阶复系数对称微分算子自共轭域（尽管这里微分方程的系数是复值的）,并通过自共轭域的刻画，给出了用微分方程实参数解的性质刻画谱是离散的条件；使用全新的方法，给出了四阶自共轭边界条件新的、简明的标准型；研究了具有转移条件的耗散 Sturm-Liouville 算子以及边界条件含有谱参数的耗散奇异Dirac算子的特征值和特征函数系的完备性问题；研究了一般实系数2n阶对称微分算式在自伴的边界条件和转移条件下关于参数的连续性以及可微性，给出了特征值对参数的微分表达式；项目组还得到了具有有限多个不连续点时Sturm-Liouville算子的Weyl函数，讨论了Weyl 函数的一些性质。相关研究成果整理论文10篇在国内外期刊上发表或接收。其中，SCI索引论文6篇。