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

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

2012-04-20

arxiv.org/abs/1204.4610v1

Mathematics

Analysis of PDEs

17 pages

Scientific paper

We study the inverse problem of determining a real-valued potential in the two-dimensional Schr\"odinger 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).

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