The inverse problem of the Schr\"odinger equation in the plane; A   dissection of Bukhgeim's result
  The purpose of this licentiate thesis is to present Bukhgeim's result of 2007, which solves the inverse boundary value problem of the Schr\"odinger equation in the plane. The thesis is mainly based on Bukhgeim's paper and Kari Astala's seminar talk, which he gave the 11th and 18th September of 2008 at the University of Helsinki.   Section 3 is devoted to the history and past results concerning some related problems: notably the inverse problem of the Schr\"odinger and conductivity equations in different settings. We also describe why some of the past methods do not work in the general case in a plane domain.   Section 4 outlines Bukhgeim's result and sketches out the proof. This proof is a streamlined version of the one in Bukhgeim's paper with the stationary phase method based on Kari Astala's presentation. In the following section we prove all the needed lemmas which are combined in section 6 to prove the solvability of the inverse problem.   The idea of the proof is simple. Given two Schr\"odinger equations with the same boundary data we get an orthogonality relation for the solutions of the two equations. Then we show the existence of certain oscillating solutions and insert these into the orthogonality relation. Then by a stationary phase argument we see that the two Schr\"odinger equations are the same.   In the last section we contemplate an unclear detail in Bukhgeim's paper which Kari Astala pointed out in his seminar talk: without an extra argument Bukhgeim's proof shows the solvability of the inverse problem only for differentiable potentials instead of ones in L^p. But the special oscillating solutions exist even for L^p potentials. 
Eemeli Bl{\aa}sten
math.AP math.CV