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This result will be published as part of my PhD thesis after somestreamlining. This manuscript contains the proof of the claim, but is notpeer-reviewed.  We prove uniqueness and stability for the inverse problem of the 2DSchr\"odinger equation in the case that the potentials give well posed directproblems and are in W^{{\epsilon},p}({\Omega}), {\epsilon}>0, p>2. The idea ofthe proof is to use Bukhgeim's oscillating exponential solutions. ByAlessandrini's identity and stationary phase we get information about thedifference of the potentials from the difference of the Dirichlet-Neumann maps.  Using interpolation, we see that the the worst of the remainder terms decayswith an exponent of 1 - {\epsilon} - {\beta}. Here {\beta} is the exponentwhich we get in a norm estimate for the conjugated Cauchy operator. We get itarbitrarily close to 1, so there is uniqueness and stability when {\epsilon} >0.

Stability and uniqueness for the inverse problem of the Schr\"odinger equation in 2D with potentials in W^{{\epsilon},p}