A vectorial inverse nodal problem
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- by Yan-Hsiou Cheng, Chung-Tsun Shieh and C. K. Law PDF
- Proc. Amer. Math. Soc. 133 (2005), 1475-1484 Request permission
Abstract:
Consider the vectorial Sturm-Liouville problem: \[ \left \{\begin {array}{l} -{\mathbf y}''(x)+P(x){\mathbf y}(x) = \lambda I_{d}{\mathbf y}(x) A{\mathbf y}(0)+I_{d}{\mathbf y}’(0)={\mathbf 0} B{\mathbf y}(1)+I_{d}{\mathbf y}’(1)={\mathbf 0} \end {array}\right . \] 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.References
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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
- 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
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 1475-1484
- MSC (2000): Primary 34B24, 34C10
- DOI: https://doi.org/10.1090/S0002-9939-04-07679-8
- MathSciNet review: 2111948