Reconstruction and Stability for Piecewise Smooth Potentials in the Plane

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