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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A vectorial inverse nodal problem


Authors: Yan-Hsiou Cheng, Chung-Tsun Shieh and C. K. Law
Journal: Proc. Amer. Math. Soc. 133 (2005), 1475-1484
MSC (2000): Primary 34B24, 34C10
Published electronically: November 19, 2004
MathSciNet review: 2111948
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Abstract: Consider the vectorial Sturm-Liouville problem:

\begin{displaymath}\left\{\begin{array}{l} -{\mathbf y}''(x)+P(x){\mathbf y}(x) ... ...hbf y}(1)+I_{d}{\mathbf y}'(1)={\mathbf 0} \end{array}\right. \end{displaymath}

where $P(x)=[p_{ij}(x)]_{i,j=1}^{d}$ is a continuous symmetric matrix-valued function defined on $[0,1]$, and $A$ and $B$ are $d\times d$ real symmetric matrices. An eigenfunction ${\mathbf y}(x)$ of the above problem is said to be of type (CZ) if any isolated zero of its component is a nodal point of ${\mathbf y}(x)$. We show that when $d=2$, there are infinitely many eigenfunctions of type (CZ) if and only if $(P(x), A, B)$ are simultaneously diagonalizable. This indicates that $(P(x), A, B)$ can be reconstructed when all except a finite number of eigenfunctions are of type (CZ). The results supplement a theorem proved by Shen-Shieh (the second author) for Dirichlet boundary conditions. The proof depends on an eigenvalue estimate, which seems to be of independent interest.


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Additional Information

Yan-Hsiou Cheng
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Republic of China
Email: jengyh@math.nsysu.edu.tw

Chung-Tsun Shieh
Affiliation: Department of Mathematics, Tamkang University, Tamsui, Taipei County, Taiwan 251, Republic of China
Email: ctshieh@math.tku.edu.tw

C. K. Law
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Republic of China
Email: law@math.nsysu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07679-8
PII: S 0002-9939(04)07679-8
Received by editor(s): August 27, 2003
Received by editor(s) in revised form: February 4, 2004
Published electronically: November 19, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.