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

Abstract

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).

17 pages

References in corpus (5)

• The d-bar approach to approximate inverse scattering at fixed energy in three dimensions
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• New global stability estimates for the Gel'fand-Calderon inverse problem
• New global stability estimates for the Calderón problem in two dimensions
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Cited by in corpus (4)

• Effectivized Holder-logarithmic stability estimates for the Gel'fand inverse problem
• Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions
• The Born approximation for the fixed energy Calderón problem
• Increasing resolution and instability for linear inverse scattering problems