| 郭永霞,杨传富,黄振友.具有特征参数多项式边界条件的Sturm-Liouville 方程的逆结点问题[J].数学年刊A辑,2012,33(6):705~718 |
| 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 |
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| DOI： |
| 中文关键词: Sturm-Liouville方程, 参数边界条件, 逆结点问题, 势函数 |
| 英文关键词:Sturm-Liouville equation, Parameter boundary condition,
Inverse nodal problem, Potential function |
| 基金项目:国家自然科学基金(No.11171152)和江苏省自然科学基金(No.BK2010489) |
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| 中文摘要: |
| 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 $[0,b]$ \big($ b\in \big(\frac{1}{2},1\big]$\big) 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. |
| 英文摘要: |
| Inverse nodal problem consists in constructing operators from the given nodes of their eigenfunctions.
In this paper, inverse nodal problems for the Sturm-Liouville problem 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 conditions 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 $(0,b)$ \big(for any fixed $ b \in \big(\frac{1}{2},1\big]$\big) determines the potential on $ [0,1]$
and the unknown coefficient of polynomials in the boundary conditions. |
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