Recovering differential pencils with spectral boundary conditions and spectral jump conditions

[摘要] 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: $(i)$ the potentials $q_{k}(x)$ and boundary conditions of such a problem can be uniquely established by some information on eigenfunctions at some internal point $b\in (\frac{\pi }{2},\pi )$ and parts of two spectra; $(ii)$ if one boundary condition and the potentials $q_{k}(x)$ are prescribed on the interval $[\pi /2(1-\alpha ),\pi ]$ for some $\alpha \in (0, 1)$ , then parts of spectra $S\subseteq \sigma (L)$ are enough to determine the potentials $q_{k}(x)$ on the whole interval $[0, \pi ]$ and another boundary condition.

[学科分类] 电力

[关键词] Inverse problem;Differential pencil;Spectral boundary condition;Spectral jump condition