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Discontinuous Sturm-Liouville problems containing eigenparameter in the boundary conditions. (English) Zbl 1112.34070

The authors consider the following discontinuous Sturm-Liouville eigenvalue problems with eigenvalue parameters both in the equation and in one of the boundary conditions:

$\tau u:=-a\left(x\right){u}^{\text{'}\text{'}}+q\left(x\right)u=\lambda u,x\in \left[-1,0\right)\cup \left(0,1\right]$

with the boundary conditions at $x=±1$

${L}_{1}u:={\alpha }_{1}u\left(-1\right)+{\alpha }_{2}{u}^{\text{'}}\left(-1\right)=0;{L}_{4}\left(\lambda \right)u:=\lambda \left({\beta }_{1}^{\text{'}}u\left(1\right)-{\beta }_{2}^{\text{'}}{u}^{\text{'}}\left(1\right)\right)+\left(\left({\beta }_{1}u\left(1\right)-{\beta }_{2}{u}^{\text{'}}\left(1\right)\right)=0,$

and the transition conditions

${L}_{2}u:={\gamma }_{1}u\left(-0\right)-{\delta }_{1}u\left(+0\right)=0,{L}_{3}u:={\gamma }_{2}{u}^{\text{'}}\left(-0\right)-{\delta }_{2}{u}^{\text{'}}\left(+0\right)=0,$

where $a\left(x\right)={a}_{1}^{2}>0$ for $x\in \left[-1,0\right)$, $a\left(x\right)={a}_{2}^{2}>0$ for $x\in \left(0,1\right]$; $q\left(x\right)$ is a given real-valued function which is continuous in $\left[-1,0\right]$ and in $\left[0,1\right]$; ${\alpha }_{i},{\beta }_{i},{\beta }_{i}^{\text{'}},{\gamma }_{i},{\delta }_{i},i=1,2$, are real numbers with $|{\alpha }_{1}|+|{\alpha }_{2}|\ne 0,|{\beta }_{1}^{\text{'}}|+|{\beta }_{2}^{\text{'}}|+|{\beta }_{1}|+|{\beta }_{2}|\ne 0,|{\gamma }_{1}|+|{\delta }_{1}|\ne 0,|{\gamma }_{2}|+|{\delta }_{2}|\ne 0$, and ${\beta }_{1}^{\text{'}}{\beta }_{2}-{\beta }_{1}{\beta }_{2}^{\text{'}}>0$. With an operator theoretic approach, some classical properties and asymptotic approximate formulae for eigenvalues and normalized eigenfunctions are obtained.

##### MSC:
 34L20 Asymptotic distribution of eigenvalues for OD operators 34B24 Sturm-Liouville theory
##### References:
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