基于量子力学的算子谱理论问题研究

In quantum mechanics, energy operator is a self-adjoint operator defined on some space, whose eigenvalue corresponds to the system level of the bound state, while the spectrum is the eigenvalue distribution of an operator. And the eigenvalue distribution of the operator is involved in evaluating the stablility and frequency of the system etc. Spectral theory, therefore, must be one of the most important mathematical theories in quantum mechanics. This research gives a further study on some important theoretical problems related on the operator spectrum theory in the quantum mechanics. The key points of the research are: the distribution of the eigenvalue problem and the problem of operator circularity. The eigenvalue distribution problem in the research mainly focus on : (1) The characteristic of the new variants of Weyl type theorem; (2) The study of the relationship of Weyl type theorem and its variants; (3) Preservation of the variants of Weyl type theorem under compact perturbation. The variants of Weyl type theorem is considered in a new way, and is connected with the cyclicity of the operator and the consistency of the spectrum property. The research not only offers an essential mathematical foundation for the research of quantum mechanics, but also provides new aspects on the study of operator algebra and operator theory, aiming to promote the development of the insection of the two disciplines.

在量子力学中，能量算符是某一空间的一个自伴算子，其本征值对应着该系统束缚态的能级，而光谱为某个算子本征值的分布，判定系统的稳定性、求系统的频率等均涉及到算子本征值的分布问题。因此，谱理论必然是量子力学中最重要的数学理论基础之一。本项目将深入研究量子力学中提出的有关算子谱理论方面的若干重要理论问题。重点研究：本征值的分布问题和算子循环性问题。其中本征值分布问题中重点解决以下几个问题：(1) Weyl型定理新的变化性质的判定。(2) Weyl型定理及其各种变化性质之间关系的研究。(3) Weyl型定理变化性质在紧摄动下的保持问题。从新的角度考虑了Weyl型定理的变化性质，并将该问题和算子循环性及谱集的一致性质结合起来，该研究不仅为量子力学的研究奠定必要的数学理论基础，也将为算子理论与算子代数的发展提出新的研究问题，注入新的活力，促进两个学科之间的交叉发展。

受国家自然科学基金委的资助，按照研究计划，对量子力学中提出的有关算子谱理论方面的问题进行了研究。研究了以下问题：(1) Weyl 型定理变化性质的判定；(2) Weyl 型定理变化性质与亚循环性质之间的关系；(3) Weyl 型定理变化性质稳定性的判定。对这三个问题的研究细致深入，得到了较深的结果。部分结果发表或者待发表于国内外数学期刊，目前已经在国内外杂志上发表标注基金号的项目23篇，其中在SCI期刊上发表文章3篇(含录用待发表1篇)，在CSCD核心期刊上发表文章7篇。