A vectorial inverse nodal problem

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Title:	A vectorial inverse nodal problem
Authors:	Cheng, Yan-hsiou;Shieh, Chung-tsun;謝忠村;Law, C. K.
Contributors:	淡江大學數學學系
Date:	2005-05
Issue Date:	2010-01-28 07:44:02 (UTC+8)
Publisher:	Providence: American Mathematical Society (AMS)
Abstract:	Consider the vectorial Sturm-Liouville problem: −y''(x) + P(x)y(x) = λIdy(x) Ay(0) + Idy(0) = 0 By(1) + Idy(1) = 0 where P(x)=[pij(x)]d i,j=1 is a continuous symmetric matrix-valued function defined on [0, 1], and A and B are d×d real symmetric matrices. An eigenfunction y(x) of the above problem is said to be of type (CZ) if any isolated zero of its component is a nodal point of y(x). We show that when d = 2, there are infinitely many eigenfunctions of type (CZ) if and only if (P(x), A, B) are simultaneously diagonalizable. This indicates that (P(x), A, B) can be reconstructed when all except a finite number of eigenfunctions are of type (CZ). The results supplement a theorem proved by Shen-Shieh (the second author) for Dirichlet boundary conditions. The proof depends on an eigenvalue estimate, which seems to be of independent interest.
Relation:	Proceedings of the American Mathematical Society 133(5), pp.1475-1484
DOI:	10.1090/S0002-9939-04-07679-8
Appears in Collections:	[Graduate Institute & Department of Mathematics] Journal Article

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