In this paper we investigate the uniqueness of potential recovery in the inverse Borg–Levinson problem, and our study deals with a model of its recovery. The article applies the theory of elliptic equations. Using the resolvent method, in fact, we obtain uniqueness conditions for potential recovery in the inverse Borg–Levinson problem with Newton’s boundary condition considered on a multidimensional parallelepiped if the nature of the asymptotic decomposition of eigenvalues is known. The inverse Borg–Levinson problem with Dirichlet’s boundary condition was investigated in Sadovnichii et al. (Dokl Math 61(1):67–69, 2000). It was proved that the lack of knowledge about a finite number of spectral data does not affect the uniqueness of the potential recovery, and we can also exclude some infinite sequences of eigenvalues that have certain properties, which will not affect the uniqueness of the potential recovery. Here the authors consider a problem with more general Newton’s boundary condition. The study convinces that with a certain asymptotic behavior of the eigenvalues, which is valid on a two-dimensional and three-dimensional cube and cannot be considered on it as an additional requirement, with the eigenfunctions bounded at the boundary and some other conditions, a uniqueness result for the potential restoration holds true. This result is also valid for a problem with Neumann’s boundary condition. A potential recovery model was formulated. If the potential of one of their tasks is known, and the difference between the spectral characteristics of the other problem with an unknown potential and the spectral characteristics of the initial one satisfies the presented conditions, then the potentials of these problems coincide.

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关于Borg-Levinson反问题中的数学势能恢复模型问题的一些说明

在本文中，我们研究了Borg-Levinson反问题中潜在恢复的唯一性，我们的研究涉及其恢复模型。本文应用了椭圆方程的理论。实际上，使用分解方法，如果已知特征值的渐近分解的性质，则可以在多维平行六面体上考虑牛顿边界条件，获得逆Borg-Levinson问题中潜在恢复的唯一性条件。Sadovnichii等人研究了Dirichlet边界条件下的Borg-Levinson逆问题。（Dokl Math 61（1）：67-69，2000）。事实证明，对有限数量的光谱数据缺乏了解不会影响潜在恢复的唯一性，而且我们还可以排除某些具有某些特性的特征值的无限序列，这不会影响潜在恢复的唯一性。在这里，作者考虑了更一般的牛顿边界条件的问题。该研究认为，本征值具有一定的渐近性，在二维和三维立方体上有效，并且不能将其视为附加要求，本征函数在边界和其他一些条件下是有界的。潜在恢复的唯一性结果成立。这个结果对于诺伊曼边界条件的问题也是有效的。建立了潜在的恢复模型。如果他们的一项任务的潜力是已知的，而另一问题的光谱特征与未知的电位之间的差异与最初任务的光谱特征之间的差异满足所提出的条件，