厄密线性关系扰动理论及其在奇异哈密顿系统中的应用

The classical theory of linear operators cannot be applied to define the adjoint of a non-densely defined operator and the inverse of a non-injective operator. In order to make up the deficiency, von Neumann first introduced linear relations into functional analysis and Arens futher studied and developed the theory of linear relations. Since the theory of linear relations was established, the perturbation problems of linear relations have appeared in the field of engineering technology, fluid mechanics, and quantum mechanics etc. These problems have attracted extensive attention of scholars both at home and abroad. Among these problems, the perturbation problem of Hermitian linear relations is a hot research topic. In the present project, we will apply the theories of linear relations and functional analysis to carry out the study of perturbation theory for Hermitian linear relations. It contains mainly the equivalent characterization and criterion of various perturbations for linear relations, stability and change estimation of deficiency indices of Hermitian linear relations under relatively bounded, relatively compact, and gap perturbations, stability and change estimation of absolutely continuous spectra of self-adjoint linear relations under trace class perturbations, application to discrete linear Hamiltonian systems, generalization to singular Hamiltonian systems under perturbation on time scales. It is believed that the results of this project can not only make complete the perturbation theory for Hermitian linear relations, but also provide the theoretical premise and foundation for the study of self-adjoint extensions and distribution of spectrum of dynamic systems in the discrete, continuous, and time scale cases.

为了弥补经典线性算子理论无法定义非稠定算子的伴随和非单射算子的逆的不足，von Neumann最早在泛函分析中引入线性关系。Arens对线性关系理论进行了深入研究并推动了该理论的发展。自线性关系理论建立以来，线性关系扰动问题广泛出现在工程技术、流体力学、量子力学等领域中。该问题的研究引起了国内外学者的广泛关注，其中厄密线性关系的扰动问题是一个研究热点。本项目拟利用线性关系理论和泛函分析理论，研究厄密线性关系扰动理论，主要包括线性关系各类扰动的等价刻画和判定；厄密线性关系亏指数在相对有界、相对紧以及间隙度量意义扰动下的稳定性及其变化估计；自伴线性关系绝对连续谱在迹类扰动下的稳定性和变化估计及其判定定理；最后，将上述结果应用于离散哈密顿系统和推广到时间尺度上奇异哈密顿系统中。本项目研究将进一步完善厄密线性关系扰动理论，为研究离散、连续以及时间尺度上的动力系统自伴扩张和谱分布提供理论前提和基础。

为了弥补经典线性算子理论无法定义非稠定算子的伴随和非单射算子的逆的不足，von Neumann最早在泛函分析中引入线性关系。Arens对线性关系理论进行了深入研究并推动了该理论的发展。自线性关系理论建立以来，线性关系扰动问题广泛出现在工程技术、流体力学、量子力学等领域中。本项目主要研究了厄密线性关系扰动理论，并将该理论应用于研究奇异二阶差分方程和奇异离散哈密顿系统的相关问题，主要包括闭线性关系迹类扰动的等价刻画；厄密线性关系的亏指数在相对有界扰动下的稳定性；奇异二阶对称线性差分方程和奇异离散哈密顿系统的亏指数在相对有界扰动下的稳定性；自伴线性关系的绝对连续谱在迹类扰动和有限秩扰动下的稳定性。该项目所获得的结果发展了线性关系扰动理论，将为研究离散、连续以及时间尺度上的动力系统自伴扩张和谱分布提供理论前提和基础。

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