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Reconstruction and stability for piecewise smooth potentials in the plane

We show that potentials with jump discontinuities can be recovered from the Dirichlet-to-Neumann map using Bukhgeim's method. Combining with known formulas, this enables the recovery from the scattering amplitude at a fixed energy. We also provide a priori stability estimates for reconstruction from the Dirichlet-to-Neumann map as well as from the scattering amplitudegiven an approximate knowledge of the location of the discontinuities.