运用辛几何方法研究自共轭算子与耗散算子的定义域和谱分析

As an important geometric method for the study of differential operator theory, symplectic geometry is very effective in characterizing the expansion of symmetric operators and has a wide range of generality. In this projection, the method of symplectic geometry will be extended to the study of spectral theory, and further refinement of the characterizations of the extensions is the premise and basis for the study of spectral theory. We intend to explore the self-adjoint extensions and related spectral problem by symplectic geometry: Characterize the Cayley transform by symplectic geometry; Describe the conjugate operator of symmetric operator; Study the influence on the number, multiple numbers, and distribution of the eigenvalues caused by self-adjoint extensions, then reveal the "wonderful" role of the minimal domain. On this basis, the work is extended to the study of dissipative extensions: Study the essential differences, especially the connections and differences of the eigenvalue distributions and the eigenfunctions between the non strictly dissipative extensions and strictly dissipative extensions. Investigate the general characterization of Friedrichs extensions and the lower bound of the discrete spectrum and the essential spectrum by combining symplectic geometry and limit-circle solutions. Using balanced intersection principle, we will study the canonical forms of complete Lagrangian subspaces and illustrate the influence of different canonical forms on the eigenvalue distribution of the corresponding self-adjoint operators. This research is expected to lead to more new questions, It is hoped to explore a new approach to the study of the spectral theory.

辛几何作为研究微分算子理论的重要几何方法，在刻画对称算子的扩张方面非常有效且具有广泛的一般性。课题拟将辛几何方法延伸到微分算子谱理论的研究，而进一步完善各类扩张的刻画是研究谱理论的前提和基础。我们拟利用辛几何探讨自共轭扩张及相关谱问题：从辛几何角度刻画Cayley变换；给出对称算子的共轭算子的辛几何描述；研究自共轭扩张对特征值个数、重数、分布的影响，揭示最小算子域起到的“奇妙”作用。在此基础上把工作拓展到耗散算子的研究：利用辛几何探索严格和非严格耗散扩张有哪些本质的区别？两者的特征值分布与特征函数有什么联系和不同？结合辛几何和实参数极限圆解来研究Friedrichs扩张的一般刻画及其离散谱和本质谱的下界。利用平衡相交原理考虑完全Lagrangian子空间的标准型以及不同标准型对相应自共轭算子特征值分布的影响。该研究预期引发更多新的研究问题，希望为谱理论探讨一条新的研究途径。

辛几何是微分算子理论的重要研究方法，课题组主要利用辛几何的方法围绕对称微分算子的自共轭扩张与耗散扩张展开了多项研究工作：复辛空间中耗散子空间的构造与分类；复辛空间中完全Lagrangian子空间的构造与标准型；对称算子的对称扩张算子耗散扩张算子及其共轭算子的研究；此外在具有含参边条件的不连续Sturm-Liouville 算子以及Friedrichs 扩张方面做了多项研究工作。重要结果包括：对称扩张算子的共轭算子收缩的元素是Lagrangian元素；自共轭算子作为最大算子的收缩，收缩掉的元素构成另一个自共轭算子，称为原自共轭算子的对偶自共轭算子；进一步证明了对偶自共轭算子的不唯一性；给出复辛空间的子空间是完全Lagrangian子空间的充要条件，并揭示了它与Cayley 变换以及自共轭边界条件的内在统一性。探讨了自共轭算子与对偶耗散算子的特征值之间的联系以及严格耗散算子与非严格耗散算子点谱的区别。本课题不仅完善了复辛空间理论且初步利用辛几何方法探究了微分算子的谱性质。