奇异J-对称哈密顿系统谱问题研究

项目名称： 奇异J-对称哈密顿系统谱问题研究

项目编号： No.11471191

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 孙华清

作者单位： 山东大学

项目金额： 65万元

中文摘要： 微分算子谱理论分为对称与非对称两种。经典力学是对称力学，宏观牛顿力学理论体系和经典量子力学理论体系所产生的大量微分算子是对称的，而许多物理试验和自然现象中所呈现的规律一般是非对称的。在这些非对称问题中有相当一部分是J-对称的，这自然要求我们对J-对称微分算子谱性质进行全面地了解和掌握。本项目将对奇异J-对称哈密顿系统的亏指数类型、J-自伴扩张、相应算子增生条件与扇形条件以及相应J-自伴算子谱性质进行深入地研究。将建立亏指数类型判别准则，给出系统最小算子所有J-自伴扩张刻画，给出算子增生条件与扇形条件。特别地，本项目注重方法的使用，将利用系统系数给出本质谱不变的扰动条件，给出相应算子谱与平方可积解之间关系，并利用所获结果结合对称转化法以及奇异列法等研究谱性质。本项目拟完成的成果对于数学理论本身、原子核物理、电磁场理论、量子力学以及许多科学领域的研究提供新思路新方法。

中文关键词： 哈密顿系统；J-对称；亏指数；J-自伴扩张；谱

英文摘要： The spectral theory for differential operators can be classified into the symmetric case and the non-symmetric case. The classical mechanics is symmetric. The theory for macroscopical Newton mechanics and that for classical quantum mechanics are symmetric. However, the laws of many physical tests and natural phenomena are non-symmetric in general, and a lot of them are J-symmetric. Therefore, we must comprehensively understand the properties of J-symmetric differential operators. In this project, we will consider singular J-symmetric Hamiltonian systems, and study the defect index types of J-symmetric Hamiltonian systems, the characterizations of all the J-self-adjoint extensions of the minimal operator, and the conditions for the corresponding operators to be accretive and sectorial. In particular, we will pay attention to the methods applied. The conditions for the essential spectrum to be invariant will be given, and the relationships between the spectrum of the corresponding operator and the integrable square solutions will be obtained. By using the above results, the techniques that the non-symmetric differential systems are transformed into a pencil of symmetric systems, and the singular sequence theory, the spectral properties will be studied. The results of this project that will be obtained are mew theories and methods for us to study the mathematics itself, nuclear physics, electromagnetic field theory, and quantum mechanics etc.

英文关键词： Hamiltonian system;J-symmetric;Defect index;J-self-adjoint extension;Spectrum