Recovering differential pencils with spectral boundary conditions and spectral jump conditions

• Main Authors: Yasser Khalili, Dumitru Baleanu
• Format: Article
• Language: English
• Published: SpringerOpen 2020-12-01
• Series: Journal of Inequalities and Applications
• Subjects: Inverse problem; Differential pencil; Spectral boundary condition; Spectral jump condition
• Online Access: https://doi.org/10.1186/s13660-020-02537-z

Abstract In this work, we discuss the inverse problem for second order differential pencils with boundary and jump conditions dependent on the spectral parameter. We establish the following uniqueness theorems: ( i ) $(i)$ the potentials q k ( x ) $q_{k}(x)$ and boundary conditions of such a problem can be uniquely established by some information on eigenfunctions at some internal point b ∈ ( π 2 , π ) $b\in (\frac{\pi }{2},\pi )$ and parts of two spectra; ( i i ) $(ii)$ if one boundary condition and the potentials q k ( x ) $q_{k}(x)$ are prescribed on the interval [ π / 2 ( 1 − α ) , π ] $[\pi /2(1-\alpha ),\pi ]$ for some α ∈ ( 0 , 1 ) $\alpha \in (0, 1)$ , then parts of spectra S ⊆ σ ( L ) $S\subseteq \sigma (L)$ are enough to determine the potentials q k ( x ) $q_{k}(x)$ on the whole interval [ 0 , π ] $[0, \pi ]$ and another boundary condition.