Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài :Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval | Chang and Shieh Boundary Value Problems 2011 2011 40 http content 2011 1 40 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval Tsorng-Hwa Chang1 2 and Chung-Tsun Shieh 1 Correspondence ctshieh@mail. department of Mathematics Tamkang University Yingzhuan Rd. Danshui Dist. New Taipei City 25137 Taiwan PR China Full list of author information is available at the end of the article Springer Abstract In this paper the vectorial Sturm-Liouville operator Lq d Q x is considered dx2 where Q x is an integrable m X m matrix-valued function defined on the interval 0 n 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 m m 1 1 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 1. Introduction The study on inverse spectral problems for the vectorial Sturm-Liouville differential equation y XIm Q x y 0 0 x n on a finite interval is devoted to determine the potential matrix Q x from the spectral data of with boundary conditions U y y 0 hy 0 0 V y y n Hy n 0 where l is the spectral parameter h hij i j i-m and H Hij j j ĩ m are in Mn C and Q x Qij x i j i m is an integrable matrix-valued function. We use Lm L Q h H to denote the boundary problem - . For the case m 1 - is a scalar Sturm-Liouville equation. The scalar Sturm-Liouville equation often arises from some physical problems for example vibration of a string quantum mechanics and geophysics. Numerous research results for this case have been established by renowned mathematicians notably Borg Gelfand .
