  This result will be published as part of my PhD thesis after some
streamlining. This manuscript contains the proof of the claim, but is not
peer-reviewed.
  We prove uniqueness and stability for the inverse problem of the 2D
Schr\"odinger equation in the case that the potentials give well posed direct
problems and are in W^{{\epsilon},p}({\Omega}), {\epsilon}>0, p>2. The idea of
the proof is to use Bukhgeim's oscillating exponential solutions. By
Alessandrini's identity and stationary phase we get information about the
difference of the potentials from the difference of the Dirichlet-Neumann maps.
  Using interpolation, we see that the the worst of the remainder terms decays
with an exponent of 1 - {\epsilon} - {\beta}. Here {\beta} is the exponent
which we get in a norm estimate for the conjugated Cauchy operator. We get it
arbitrarily close to 1, so there is uniqueness and stability when {\epsilon} >
0.
