微分算子自伴域的刻画及谱的离散性

We will discuss the spectrum analysis, inverse problem and asymptotic formulas of eigenvalues of discontinuous differential operators;　The characterization of self-adjoint domain of differential operators on multi-intervals and products of differential expressions will be obtained in terms of real-parameter solutions. Discreteness of spectrum of self-adjoint and J-self-adjoint differential operators will be studied by means of the operator decomposition and quadratic comparison method. Also numerical calculations of Sturm-Liouville problems on multi-intervals will be investigated . Our purpose is to study the spectrum of differential operators in many ways such　as the　boundary conditions and the transmission conditions, the real-parameter solutions, the coefficients of differential expressions , the first class of numerical calculations etc.. Their research value is to thoroughly reveal the relationship between the spectrum of differential operators and the ways mentioned above. It's worth noting that it is widespread concern that differential operator be studied with of boundary conditions and transmission conditions and the real parameter solutions in international differential operators field.

本项目拟研究不连续微分算子的谱分析与反问题、特征值的渐进估计; 利用实参数解刻画多区间上微分算子的自伴域与微分算式乘积的自伴域; 利用算子分解的方法和二次型比较的方法来讨论自伴和J-自伴微分算子谱的离散性; 研究多区间上Sturm-Liouville问题中第一类数值计算等．其目标是从边界条件和转移条件、实参数解、微分算式的系数、第一类数值计算等多角度对微分算子的谱展开研究，其研究价值在于深入揭示微分算子的谱与边界条件和转移条件、实参数解、微分算式系数的关系. 利用边界条件和转移条件、实参数解来研究微分算子的谱是国际上本领域一个全新的和广泛关注的研究方向．

本项目研究了不连续微分算子的谱分析与反问题、特征值的渐进估计；利用微分算子谱理论及函数论的方法研究了特征值对边界的依赖性；利用转移条件定义新的内积,结合紧算子的谱理论以及逆算子的相关性质,研究了边界条件含有特征参数的高阶不连续微分算子特征函数系的完备性；利用实参数解刻画多区间上微分算子的自伴域与微分算式乘积的自伴域；利用算子分解的方法和二次型比较的方法讨论了系数中带有幂函数和指数函数的高阶对称微分算子谱的离散性；利用1-谱族特征值不等式研究了数值求解特征值的下标问题.达到了从边界条件和转移条件、实参数解、微分算式的系数、数值计算等多角度对微分算子的谱展开研究的目标.