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

DOI:
https://doi.org/10.1090/S0002-9939-99-05031-5

Published electronically:
April 28, 1999

MathSciNet review:
1622977

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that under certain conditions, a vectorial Sturm-Liouville differential equation of dimension can only possess finitely many eigenvalues which have multiplicity . For the case , we find a sufficient condition on the potential function , and a bound depending on , such that the eigenvalues of the equation with index exceeding 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.

**1.**Z. S. Agranovich and V. A. Marchenko,*The Inverse Problem of Scattering Theory*, Gordon and Breach, New York, 1963. MR**28:5696****2.**F. V. Atkinson,*Discrete and Continuous Boundary Problems*, Academic Press, New York, 1964. MR**31:416****3.**E. Hille,*Lectures on Ordinary Differential Equations*, Addison-Wesley, Massachusetts, 1969. MR**40:2939****4.**G. P. Tolstov,*Fourier Series*, Dover, New York, 1976. MR**54:13429****5.**G. N. Watson,*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, Cambridge, 1980. MR**96i:33010****6.**R. Courant and D. Hilbert,*Methods of Mathematical Physics, Vol. I*, Interscience, New York, 1953. MR**16:426a****7.**I. C. Gohberg and M. G. Krein,*Theory an Applications of Volterra Operators in Hilbert Space, Translations of Mathematical Monographs, Vol. 24*, AMS, 1970. MR**41:9041**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
34A30,
34B24,
34B25,
34L05

Retrieve articles in all journals with MSC (1991): 34A30, 34B24, 34B25, 34L05

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:
https://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