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On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equation and some related spectral problems

Author(s): Chao-Liang Shen; Chung-Tsun Shieh
Journal: Proc. Amer. Math. Soc. 127 (1999), 2943-2952.
MSC (1991): Primary 34A30, 34B24, 34B25, 34L05
Posted: April 28, 1999
MathSciNet review: 1622977
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Abstract: We prove that under certain conditions, a vectorial Sturm-Liouville differential equation of dimension $\begin{math}n \geq 2 \end{math}$ can only possess finitely many eigenvalues which have multiplicity $\begin{math}n\end{math}$ . For the case $\begin{math}n=2\end{math}$ , we find a sufficient condition on the potential function $\begin{math}Q(x)\end{math}$ , and a bound $\begin{math}m_Q\end{math}$ depending on $\begin{math}Q(x)\end{math}$ , such that the eigenvalues of the equation with index exceeding $\begin{math}m_Q\end{math}$ are all simple. These results are applied to find some sufficient conditions which imply that the spectra of two potential equations, or two string equations, have finitely many elements in common, and an estimate of the number of elements in the intersection of two spectra is provided.

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Additional Information:

Chao-Liang Shen
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Email: shen@math.nthu.edu.tw

Chung-Tsun Shieh
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Address at time of publication: Department of Mathematics, Fu Jen Catholic University, Hsinchuang, Taipei, Taiwan 24205, Republic of China
Email: ctshieh@math.fju.edu.tw

DOI: 10.1090/S0002-9939-99-05031-5
PII: S 0002-9939(99)05031-5
Keywords: Vectorial Sturm-Liouville differential equations, eigenvalues, multiplicity, spectrum, potential equations, string equations, Bessel functions
Received by editor(s): December 29, 1997
Posted: April 28, 1999
Communicated by: Hal L. Smith
Copyright of article: Copyright 1999, American Mathematical Society

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