Stability and uniqueness for the inverse problem of the Schrödinger equation in 2D with potentials in W^{ε,p}

Eemeli Blåsten

Published 2011-06-03Version 1

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 This manuscript contains the proof of the claim, but is not peer-reviewed.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.