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Eigenvalue problems for Sturm-Liouville equations with transmission conditions. (English) Zbl 1073.34096
For the Sturm-Liouville eigenvalue problem $-{u}^{\text{'}\text{'}}+q\left(x\right)u=\lambda u$ for $x\in \left(-1,1\right),x\ne 0$ subject to the eigenparameter-dependent boundary conditions $u\left(-1\right)=0,\left(\lambda -a\right){u}^{\text{'}}\left(1\right)+\lambda bu\left(1\right)=0$ and the transmission conditions at $x=0$: $u\left(-0\right)={h}_{1}u\left(+0\right),{u}^{\text{'}}\left(-0\right)={h}_{2}{u}^{\text{'}}\left(+0\right)$, 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 $\left[-1,0\right)\cup \left(0,1\right]$ 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:
 34L20 Asymptotic distribution of eigenvalues for OD operators 34B24 Sturm-Liouville theory
##### References:
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