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Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval

Tsorng-Hwa Chang12 and Chung-Tsun Shieh1*

Author Affiliations

1 Department of Mathematics, Tamkang University, No.151, Yingzhuan Rd., Danshui Dist., New Taipei City 25137, Taiwan, PR China

2 Department of Electronic Engineering, China University of Science and Technology, No.245, Academia Rd., Sec. 3, Nangang District, Taipei City 115, Taiwan, PR China

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Boundary Value Problems 2011, 2011:40  doi:10.1186/1687-2770-2011-40

Published: 26 October 2011

Abstract

In this paper, the vectorial Sturm-Liouville operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M1">View MathML</a> is considered, where Q(x) is an integrable m × m matrix-valued function defined on the interval [0,π] The authors prove that m2+1 characteristic functions can determine the potential function of a vectorial Sturm-Liouville operator uniquely. In particular, if Q(x) is real symmetric, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M2">View MathML</a> characteristic functions can determine the potential function uniquely. Moreover, if only the spectral data of self-adjoint problems are considered, then m2 + 1 spectral data can determine Q(x) uniquely.

Keywords:
Inverse spectral problems; Sturm-Liouville equation