Proceedings of the American Mathematical Society

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

 

 

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


Authors: Chao-Liang Shen and Chung-Tsun Shieh
Journal: Proc. Amer. Math. Soc. 127 (1999), 2943-2952
MSC (1991): Primary 34A30, 34B24, 34B25, 34L05
Published electronically: April 28, 1999
MathSciNet review: 1622977
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Z. S. Agranovich and V. A. Marchenko, The inverse problem of scattering theory, Translated from the Russian by B. D. Seckler, Gordon and Breach Science Publishers, New York-London, 1963. MR 0162497
  • 2. F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
  • 3. Einar Hille, Lectures on ordinary differential equations, Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0249698
  • 4. Georgi P. Tolstov, Fourier series, Dover Publications, Inc., New York, 1976. Second English translation; Translated from the Russian and with a preface by Richard A. Silverman. MR 0425474
  • 5. G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
  • 6. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • 7. I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. MR 0264447

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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: http://dx.doi.org/10.1090/S0002-9939-99-05031-5
Keywords: Vectorial Sturm-Liouville differential equations, eigenvalues, multiplicity, spectrum, potential equations, string equations, Bessel functions
Received by editor(s): December 29, 1997
Published electronically: April 28, 1999
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society