线性算子的谱结构及其扰动分析

In quantum mechanics, the energy operator is a self adjoint operator on some space, its eigenvalue corresponds to the energy levels of the bound states of the system, and the lignt spectral is a characteristic value distribution of operator. The frequency of the system and the stability of system are related to the distribution of the eigenvalues of the operators. Therefore, spectral theory is one of the most important mathematical foundation of physics, quantum mechanics and so on. In this project, by synthetically using the thought and method of analysis, algebra and topology,the struction of spectrum for linear operators and its perturbations are discussed systematically and thoroughly in terms of operator spectrum decomposition and linear maps. We mainly study: the spectral analysis and its perturbation analysis for functions of operators;the judgement of hypercyclic property and the extension and the application of Jacobson theorem; the characterization of linear maps preserving some kinds of spectrum sets; the struction of local spectrum for quantum operation; and we will establish fixed point theory for quantum channel and quantum operation,further consider the relation between the von Neuman algebra generated by noise operators of quantum operation and its dual operation.Through the study of this project, we will further enrich the mathematical foundation in the quantum information theory, also we will establish mathematical model and provide a new research method for quantum error correction theory. This study will promote interdisciplinary.

量子力学中，能量算符是某Hilbert空间上的一个自伴算子，其特征值对应着该系统束缚态的能级，而光谱是某个算子特征值的分布，求系统的频率、判定系统的稳定性等均涉及到算子的特征值的分布问题。因此，谱理论是物理学、量子力学等学科的最重要的数学基础之一。本项目综合运用分析、代数、拓扑的思想方法，以算子谱分解及线性映射为工具，对算子的谱结构及其扰动进行研究。拟研究以下问题：(1)算子函数的谱分析及其扰动分析；(2)算子循环性的判定以及Jacobson定理的推广和应用；(3)保持各类谱集的稳定性的线性映射的刻画；(4) 量子运算的局部谱结构，建立量子运算的不动点理论,进一步考虑量子运算的噪声算子生成的von Neumann代数与对偶运算的不动点之间的联系。该研究不仅为量子力学的研究奠定必要的数学理论基础，也将为算子论和算子代数的研究提出新的问题，注入新的活力，促使两个学科之间的交叉。

受国家自然科学基金委的资助，按照研究计划，对线性算子的谱结构及其扰动进行了研究。研究了以下问题：(1)算子函数的谱分析及其扰动分析；(2)算子循环性的判定以及Jacobson定理的推广和应用；(3)保持各类谱集的稳定性的线性映射的刻画；(4) 量子运算的局部谱结构。对这四个问题的研究细致深入，得到了较为深入的结果。部分结果发表或者待发表于国内外著名数学学术期刊，目前已经在国内外杂志上发表标注基金号的文章46篇，其中在SCI期刊上发表文章19篇，在《数学学报》上发表文章3篇。