Abstract

In this paper, we investigate the inverse problem for the impulsive differential pencil in the finite interval. Taking Mochizuki–Trooshin’s theorem, it is proved that two potentials and the boundary conditions are uniquely given by one spectra together with a set of values of eigenfunctions in the situation of \(x=1/2\). Moreover, applying Gesztesy–Simon’s theorem, we demonstrate that if the potentials are assumed on the interval \([(1-\theta)/2,1],\) where \(\theta\in(0,1),\) a finite number of spectrum are enough to give potentials on \([0,1]\) and other boundary condition.

中文翻译：

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脉冲微分铅笔的反问题

摘要

在本文中，我们研究了有限区间内脉冲微分笔的反问题。利用望月-特鲁辛定理，证明了在\(x=1/2\)的情况下，两个势和边界条件由一个谱和一组本征函数值唯一给出。此外，应用 Gesztesy–Simon 定理，我们证明如果势能假设在区间\([(1-\theta)/2,1],\)上，其中\(\theta\in(0,1),\ )有限数量的谱足以给出\([0,1]\)和其他边界条件上的势。