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自伴线性关系扰动理论及其在奇异离散线性哈密顿系统中的应用

批准号11571202 学科分类定性理论与稳定性理论 ( A010702 )
项目负责人史玉明 负责人职称教授 依托单位山东大学
资助金额50.00
万元
项目类别面上项目 研究期限2016 年 01 月 01 日 至
2019 年 12 月 31 日
中文主题词多值线性算子;自伴线性关系;扰动;谱;离散线性哈密顿系统
英文主题词multivalued linear operator;self-adjoint linear relation;perturbation;spectrum;discrete linear Hamiltonian system

摘要

中文摘要 本项目拟以奇异离散线性哈密顿系统谱的扰动问题为背景,研究Hilbert空间中自伴线性关系的扰动问题,并将所获得的结果应用于奇异离散哈密顿系统谱的扰动问题之中。主要围绕以下几个课题展开研究:线性关系(即多值线性算子)的Hermite性和自伴性稳定的充分条件,以及在各种扰动(包括有界扰动、紧扰动、迹类扰动、相对有界扰动、相对紧扰动和相对迹类扰动)之下,自伴线性关系的谱及各类谱的稳定性和变化估计;这些结果在奇异离散线性哈密顿系统相关扰动问题中的应用;并将其进一步推广到更一般的时间尺度上奇异哈密顿系统的扰动问题中。由于研究的需要,我们也将关注自伴线性关系的一些基本谱问题。本项目的研究成果不仅可以发展多值线性算子理论,而且有可能改进和丰富经典算子理论中的某些结果。所获得的结果也将推动差分算子、微分算子和时间尺度上算子谱理论,以及相关应用领域的研究。
英文摘要 In this project, based on perturbation problems of spectra of singular discrete linear Hamiltonian systems, we shall study perturbation problems of self-adjoint linear relations in Hilbert spaces with applications to those of spectra of singular discrete linear Hamiltonian systems. The project will focus on the following main subjects: sufficient conditions for stability of Hermitian and self-adjoint properties of linear relations (i.e., multi-valued linear operators), and stability and error estimations of spectra and various kinds of spectra of self-adjoint linear relations under various perturbations, including bounded, compact, trace class, relatively bounded, relatively compact and relatively trace class perturbations; their applications to related perturbation problems of singular discrete linear Hamiltonian systems; and further their applications to related perturbation problems of singular linear Hamiltonian systems on time scales. Because of the study requirements, we shall also investigate some fundamental spectral problems for self-adjoint linear relations. The results obtained in this project will not only develop the theory of multi-valued linear operators, but also improve and enrich some results in the classical theory of linear operators. They will also give impetus to the spectral theory of difference operators, differential operators and operators on time scales, and the study of some related application areas.
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