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 article, 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. 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 with increasing energy (in modulus). © 2013 Copyright Taylor and Francis Group, LLC.

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

SANTACESARIA, MATTEO
2013-01-01

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 article, 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. 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 with increasing energy (in modulus). © 2013 Copyright Taylor and Francis Group, LLC.
2013
Calderón problem; generalized analytic functions; increasing stability; negative energy; Schrödinger equation; stability in 2D; Analysis; Applied Mathematics
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1018564
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