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

TL;DR

This paper proves that for the 2D Schr"odinger equation with potentials in a fractional Sobolev space, one can uniquely determine the potential from boundary measurements, with stability estimates depending on the regularity parameter.

Contribution

It establishes uniqueness and stability for the inverse Schr"odinger problem in 2D with potentials in W^{psilon,p}, extending previous results to less regular potentials.

Findings

Uniqueness of potential recovery from boundary data.

Stability estimates depending on psilon and eta.

Potential in W^{psilon,p} can be recovered with controlled error.

Abstract

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 This manuscript contains the proof of the claim, but is not peer-reviewed.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