几类微分算子特征值的可微性

发布时间：2018-10-17 10:26

【摘要】：众所周知无论在理论方面还是在应用方面,对微分算子的谱的研究都有着重要的价值,而且是一个重要的研究领域。其中,特征值问题是谱分析中的一个重要领域。本文主要围绕具有自伴边界条件的微分方程的特征值的连续可微性进行研究。首先,本文所讨论的是一类具有自伴边界条件的四阶微分方程的特征值问题。利用微分方程、泛函分析、复变函数等知识,给出了特征值和特征函数的连续性的证明。在此基础上,接着研究了特征值的可微性。借助于自伴边界条件,通过计算,我们得到了特征值关于区间端点、边界条件、方程系数和权函数的微分表达式。根据可微性的结果,得出特征值关于方程系数和权函数的单调性。其次,本文讨论了一类具有自伴边界条件的2n阶微分方程的特征值问题。证明了特征值和特征函数的连续性,并且研究了特征值的可微性,给出了特征值关于区间端点、边界条件、方程系数和权函数的微分表达式。根据本文的主要研究内容,全文可分为以下三部分:第一部分为引言,主要介绍了谱问题的研究状况和本文的主要研究结果;第二部分主要研究了自伴边界条件下的四阶谱问题的特征值的连续可微性并给出了特征值关于参数的微分表达式;第三部分主要讨论了了介绍了自伴边界条件下的2n阶谱问题的特征值的连续可微性并给出了特征值关于参数的微分表达式.
[Abstract]:It is well known that the study of the spectrum of differential operators is of great value and is an important research field both in theory and in application. Among them, eigenvalue problem is an important field in spectral analysis. This paper focuses on the continuous differentiability of eigenvalues of differential equations with self-adjoint boundary conditions. First, we discuss the eigenvalue problems of a class of fourth order differential equations with self-adjoint boundary conditions. By using the knowledge of differential equation, functional analysis and complex function, the proof of the continuity of eigenvalue and eigenfunction is given. On this basis, the differentiability of eigenvalues is studied. By means of self-adjoint boundary conditions, we obtain the differential expressions of eigenvalues for interval endpoints, boundary conditions, coefficients of equations and weight functions. According to the results of differentiability, the monotonicity of eigenvalues on the coefficients and weight functions of the equation is obtained. Secondly, we discuss the eigenvalue problems of 2n-order differential equations with self-adjoint boundary conditions. The continuity of eigenvalues and eigenfunctions is proved, and the differentiability of eigenvalues is studied. The differential expressions of eigenvalues for interval endpoints, boundary conditions, coefficients of equations and weight functions are given. According to the main research content of this paper, the full text can be divided into the following three parts: the first part is the introduction, mainly introduces the research situation of the spectrum problem and the main research results of this paper; In the second part, the continuous differentiability of the eigenvalues of the fourth order spectral problems with self-adjoint boundary conditions is studied and the differential expressions of the eigenvalues about the parameters are given. In the third part, we mainly discuss the continuous differentiability of the eigenvalues of the 2n-order spectral problems with self-adjoint boundary conditions and give the differential expressions of the eigenvalues with respect to the parameters.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.3

【相似文献】