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Eigenvalue problems for Sturm-Liouville equations with transmission conditions. (English) Zbl 1073.34096
For the Sturm-Liouville eigenvalue problem -u '' +q(x)u=λu for x(-1,1),x0 subject to the eigenparameter-dependent boundary conditions u(-1)=0,(λ-a)u ' (1)+λbu(1)=0 and the transmission conditions at x=0: u(-0)=h 1 u(+0),u ' (-0)=h 2 u ' (+0), where ab>0,h 1 h 2 >0, it was established by the authors [Appl. Math. Comput. 157, 347–355 (2004; Zbl 1060.34007)], that all eigenvalues are real and the eigenfunctions corresponding to different eigenvalues are orthogonal in a Hilbert space. In this article, the asymptotic behavior of the eigenvalues and the corresponding eigenfunctions are given through a realization of the basic solutions which are solutions of the problem on [-1,0)(0,1] except that only one boundary condition is satisfied. A counterexample is given to show that ab>0 and h 1 h 2 >0 are necessary.
MSC:
34L20Asymptotic distribution of eigenvalues for OD operators
34B24Sturm-Liouville theory
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