Carleman estimate for infinite cylindrical quantum domains and application to inverse problems

Yavar Kian (1) , Quang Sang Phan (2) , Eric Soccorsi (1)

1 CPT - Centre de Physique Théorique - UMR 7332
2 Institute of Mathematics [Krakow]

We consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain, from one Neumann boundary observation of the solution. Assuming that this potential is known outside some fixed compact subset of the waveguide, we prove that it may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schrödinger operator acting in an unbounded cylindrical domain, the Neumann data is measured on an infinitely extended subboundary of the cylinder.

Mots clés: Inverse problem; Schrödinger equation; scalar potential; Carleman estimate; infinite cylindrical domain

Domaines: Equations aux dérivées partielles [math.AP]

Yavar Kian, Quang Sang Phan, Eric Soccorsi. Carleman estimate for infinite cylindrical quantum domains and application to inverse problems. Inverse Problems, 2014, 30 (5), pp.055016. ⟨10.1088/0266-5611/30/5/055016⟩. ⟨hal-00820440v3⟩