Some notes to the issue of the mathematical potential recovery model in Borg–Levinson inverse problem

Abstract

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.