Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions

Matteo Santacesaria (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We study the inverse problem of determining a real-valued potential in the two-dimensional Schrödinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this paper we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy, for low and high energies (in modulus). In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases when increasing the energy (in modulus).

Domaines: Equations aux dérivées partielles [math.AP]

Matteo Santacesaria. Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions. Applicable Analysis, 2013, 92 (8), pp.1666-1681. ⟨10.1080/00036811.2012.698006⟩. ⟨hal-00688457v2⟩