Recovering simultaneously a potential and a point source from Cauchy data

Gang Bao (1) , Yuantong Liu (1) , Faouzi Triki (2)

1 Zhejiang University
2 EDP - Equations aux Dérivées Partielles

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.

Mots clés: inverse potential; Dirichlet to Neumann map; stability estimate; point sources; Schrödinger equation

Domaines: Equations aux dérivées partielles [math.AP]

Gang Bao, Yuantong Liu, Faouzi Triki. Recovering simultaneously a potential and a point source from Cauchy data. Minimax Theory and its Applications, 2021, 6 (2), pp.227--238. ⟨hal-03606738⟩