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.

中文翻译：

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使用光谱边界条件和光谱跳跃条件恢复差动铅笔

在这项工作中，我们讨论了具有边界和跳跃条件取决于光谱参数的二阶差分铅笔的反问题。我们建立以下唯一性定理：$（i）$势$ q_ {k}（x）$以及此类问题的边界条件可以通过某些内部点$ b \ in（\ frac {\ pi} {2}，\ pi）$和两个光谱的一部分；$（ii）$如果在某个$ \ alpha \ in区间$ [\ pi / 2（1- \ alpha），\ pi] $上规定了一个边界条件和电位$ q_ {k}（x）$ （0，1）$，则频谱$ S \ subseteq \ sigma（L）$的部分足以确定整个区间$ [0，\ pi] $和另一个区间上的电势$ q_ {k}（x）$边界条件。