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A vectorial inverse nodal problem

Authors:	Yan-Hsiou Cheng Chung-Tsun Shieh C K Law

Institution:	Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Republic of China ; Department of Mathematics, Tamkang University, Tamsui, Taipei County, Taiwan 251, Republic of China ; Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Republic of China

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 , and and 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 , there are infinitely many eigenfunctions of type (CZ) if and only if are simultaneously diagonalizable. This indicates that 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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