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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some eigenvalue problems for vectorial Sturm-Liouville equations with eigenparameter dependent boundary conditions


Author: Chi-Hua Chan
Journal: Trans. Amer. Math. Soc. 364 (2012), 119-136
MSC (2010): Primary 34B08
Published electronically: August 9, 2011
MathSciNet review: 2833579
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Abstract: We investigate the $ n$-dimensional vectorial Sturm-Liouville equation

$\displaystyle \vec{y} ''(x)+[\lambda^2 I_n -Q(x)]\vec{y}(x)=\vec{0} $

with eigenparameter dependent boundary conditions

$\displaystyle \vec{y}(0)=\vec{0}, A\vec{y} '(\pi)+\lambda \vec{y}(\pi)=\vec{0}.$

Under the assumption that $ Q(x)$ is nonnegative definite in [0,$ \pi$], we prove that the eigenvalues of the $ n$-dimensional vectorial Sturm-Liouville equation are real.

For the case $ n=2$, we show that the algebraic multiplicity of an eigenvalue of the problem as a zero of the characteristic function

$\displaystyle \omega_A(\lambda;Q)=\det [AY'(\pi;\lambda^2;Q)+\lambda Y(\pi;\lambda^2;Q)]$

is equal to its geometric multiplicity. By the theory of Hadamard's factorization, we also prove that the characteristic function $ \omega_A(\lambda;Q)$ is uniquely determined by the spectral set of the equation. Moreover, we consider the inverse problem for the equation, i.e., how many spectral sets can determine the potential function $ Q(x)$ uniquely, and we find that three spectral sets are necessary for us to determine the potential function $ Q(x)$ uniquely.


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Additional Information

Chi-Hua Chan
Affiliation: Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Email: d917201@oz.nthu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05269-4
PII: S 0002-9947(2011)05269-4
Received by editor(s): August 27, 2009
Received by editor(s) in revised form: December 8, 2009
Published electronically: August 9, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.