Minimax Theory and its Applications 06 (2021), No. 2, 227--238
Copyright Heldermann Verlag 2021
Recovering Simultaneously a Potential and a Point Source from Cauchy Data
Gang Bao
Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, P.R.China
baog@zju.edu.cn
Yuantong Liu
Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, P.R.China
ytliu@zju.edu.cn
Faouzi Triki
Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, 38401 Saint-Martin-d'Hères, France
faouzi.triki@univ-grenoble-alpes.fr
This paper is devoted to the inverse problem of recovering simultaneously a potential and a point source in a Schrödinger equation from the associated nonlinear Dirichlet to Neumann map. The uniqueness of the inversion is proved and logarithmic stability estimates are derived. It is well known that the inverse problem of determining only the potential while knowing the source, is ill-posed. In contrast the problem of identifying a point source when the potential is given is well posed. The obtained results show that the nonlinear Dirichlet to Neumann map contains enough information to determine simultaneously the potential and the point source. However recovering a point source imbedded in an unknown background medium becomes an ill-posed inversion.
Keywords: Inverse potential, Dirichlet to Neumann map, stability estimate, point sources, Schrödinger equation.
MSC: 35R30, 35C20.
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