On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition

Abstract

The spectral problem
\[-y''+q(x)y=\lambda y,\ \ \ \ 0<x<1\]
\[y(0)=0, \quad y'(0)=\lambda(ay(1)+by'(1)),\]
is considered, where $\lambda$ is a spectral parameter, $q(x)\in{{L}_{1}}(0,1)$ is a complex-valued function, $a$ and $b$ are arbitrary complex numbers which satisfy the condition $|a|+|b|\ne 0$. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in ${{L}_{p}}(0,1)$) of the above-mentioned Sturm-Liouville problem.

Keywords

• eigenvalues
• eigenfunctions
• minimal system
• basis

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