English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The purpose of this licentiate thesis is to present Bukhgeim's result of2007, which solves the inverse boundary value problem of the Schr\"odingerequation in the plane. The thesis is mainly based on Bukhgeim's paper and KariAstala's seminar talk, which he gave the 11th and 18th September of 2008 at theUniversity of Helsinki.  Section 3 is devoted to the history and past results concerning some relatedproblems: notably the inverse problem of the Schr\"odinger and conductivityequations in different settings. We also describe why some of the past methodsdo not work in the general case in a plane domain.  Section 4 outlines Bukhgeim's result and sketches out the proof. This proofis a streamlined version of the one in Bukhgeim's paper with the stationaryphase method based on Kari Astala's presentation. In the following section weprove all the needed lemmas which are combined in section 6 to prove thesolvability of the inverse problem.  The idea of the proof is simple. Given two Schr\"odinger equations with thesame boundary data we get an orthogonality relation for the solutions of thetwo equations. Then we show the existence of certain oscillating solutions andinsert these into the orthogonality relation. Then by a stationary phaseargument we see that the two Schr\"odinger equations are the same.  In the last section we contemplate an unclear detail in Bukhgeim's paperwhich Kari Astala pointed out in his seminar talk: without an extra argumentBukhgeim's proof shows the solvability of the inverse problem only fordifferentiable potentials instead of ones in L^p. But the special oscillatingsolutions exist even for L^p potentials.

The inverse problem of the Schr\"odinger equation in the plane; A dissection of Bukhgeim's result