On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficient

2012 ◽  
Vol 36 (13) ◽  
pp. 1685-1700 ◽  
Author(s):  
A. Adiloglu Nabiev ◽  
R.Kh. Amirov
Keyword(s):  
Boundary Value Problem ◽  
Liouville Equation ◽  
Boundary Value ◽  
Discontinuous Coefficient ◽  
Sturm Liouville

scholarly journals The Levinson-Type Formula for a class of Sturm-Liouville Equation

2021 ◽  
pp. 1219
Author(s):  
Sertac Goktas ◽  
Khanlar R. Mamedov
Keyword(s):  
Boundary Value Problem ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Boundary Value ◽  
Scattering Data ◽  
Scattering Function ◽  
Type Formula ◽  
Marchenko Equation ◽  
Sturm Liouville

The boundary value problem \[-{\psi}''+q(x)\psi={\lambda}^2 \psi, \quad 0<x<\infty,\] \[{\psi}'(0)-(\alpha_{0}+\alpha_{1}\lambda){\psi}(0)=0 \] is considered, where $\lambda$ is a spectral parameter, $ q(x) $ is real-valued function such that \begin{equation*} \int\limits_{0}^{\infty}(1+x)|q(x)|dx<\infty \end{equation*} with $\alpha_{0}, \alpha_{1}\geq0$ ( $\alpha_{0},\alpha_{1}\in \mathbb{R}$). In this paper, for the above-mentioned boundary value problem, the scattering data is considered and the characteristics properties (such as continuity of the scattering function $ S(\lambda) $ and giving the Levinson-type formula) of this data are studied.{\small \bf Keywords. }{Scattering data, scattering function, Gelfand-Levitan-Marchenko equation, Levinson-type formula.}

scholarly journals Some properties of eigenvalues and generalized eigenvectors of one boundary value problem

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 911-920 ◽  
Author(s):  
Hayati Olğar ◽  
Oktay Mukhtarov ◽  
Kadriye Aydemir
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Boundary Value ◽  
Discontinuous Boundary ◽  
Generalized Eigenvectors ◽  
Continuous Potential ◽  
Sturm Liouville ◽  
Piecewise Continuous

We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular, it is shown that the problem under consideration has precisely denumerable many eigenvalues ?1, ?2,..., which are real and tends to +?. Moreover, it is proven that the generalized eigenvectors form a Riesz basis of the adequate Hilbert space.

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