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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equation and some related spectral problems
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by Chao-Liang Shen and Chung-Tsun Shieh PDF
Proc. Amer. Math. Soc. 127 (1999), 2943-2952 Request permission

Abstract:

We prove that under certain conditions, a vectorial Sturm- Liouville differential equation of dimension $n \geq 2$ can only possess finitely many eigenvalues which have multiplicity $n$. For the case $n=2$, we find a sufficient condition on the potential function $Q(x)$, and a bound $m_Q$ depending on $Q(x)$, such that the eigenvalues of the equation with index exceeding $m_Q$ 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
  • Received by editor(s): December 29, 1997
  • Published electronically: April 28, 1999
  • Communicated by: Hal L. Smith
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 2943-2952
  • MSC (1991): Primary 34A30, 34B24, 34B25, 34L05
  • DOI: https://doi.org/10.1090/S0002-9939-99-05031-5
  • MathSciNet review: 1622977