具有特征参数多项式边界条件的Sturm-Liouville方程的逆结点问题  

Inverse Nodal Problems for Sturm-Liouville Equations with Boundary Conditions Depending Polynomially on the Spectral Parameter

作　　者：郭永霞[1,2] 杨传富[2] 黄振友[2] 

机构地区：[1]陕西师范大学数学与信息科学学院,西安710062 [2]南京理工大学应用数学系,南京210094

出　　处：《数学年刊（A辑）》2012年第6期705-718,共14页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.11171152);江苏省自然科学基金(No.BK2010489)资助的项目

摘　　要：逆结点问题是通过特征函数的零点重构算子.本文主要讨论具有特征参数多项式边界条件的Sturm-Liouville方程的逆结点问题.20世纪50年代以后,人们发现在许多工程领域,Sturm-Liouville问题的谱参数不仅出现在方程中,而且也出现在边界条件中,因此带参数边界条件的逆结点问题对数学物理方面的研究有重要意义.本文讨论区间[0,1]上边界条件为参数多项式的Sturm-Liouville方程的逆结点问题,并证明在[0,b](b∈(1/2,1])上结点的稠密子集可唯一确定[0,1]上的势函数和边界条件中多项式的未知系数.Inverse nodal problem consists in constructing operators from the given nodes of their eigenfunctions. In this paper, inverse nodal problems for the Sturm-Liouville prob- lem with boundary conditions depending polynomially on the spectral parameter is studied. From 1950s, mathematicians found that the spectral parameter appears not only in the equation, but also in the boundary conditions. Inverse nodal problem with boundary con- ditions depending polynomially on the spectral parameter is significant for studying the problems in mathematics and physics. The authors prove that a dense subset of the nodal set on （0, b） （for any fixed b ∈ （1/2,1]） determines the potential on [0,1] and the unknown coefficient of polynomials in the boundary conditions.

关 键 词：STURM-LIOUVILLE方程 参数边界条件 逆结点问题 势函数 

分 类 号：O175[理学—数学]