Spectral properties of Sturm-Liouville operators with discontinuities at finite pointsReport as inadecuate




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Mathematical Sciences

, 6:34

First Online: 07 September 2012Received: 11 June 2012Accepted: 23 July 2012

Abstract

PurposeIn this paper, we investigate a class of Sturm-Liouville operators with eigenparameter-dependent boundary conditions and transmission conditions at finite interior points.

MethodsBy modifying the inner product in a suitable Krein space K associated with the problem, we generate a new self-adjoint operator A such that the eigenvalues of such a problem coincide with those of A.

ResultsWe construct its fundamental solutions, get the asymptotic formulae for its eigenvalues and fundamental solutions, discuss some properties of its spectrum, and obtain its Green function and the resolvent operator.

ConclusionsThree important conclusions can be drawn: 1 the new operator A is self-adjoint in the Krein space K; 2 if θ i > 0 , i = 1 , m ¯ Open image in new window, and ρj> 0, j = 1,2, then, the eigenvalues of the problem Equations 1 to 5 are analytically simple; 3 the residual spectrum of the operator A is empty, i.e., σrA = ∅.

KeywordsSturm-Liouville operator Transmission condition Eigenparameter-dependent boundary condition 34L20 47E05  Download fulltext PDF



Author: Qiuxia Yang - Wanyi Wang

Source: https://link.springer.com/







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