The Inverse Problem of the Schrödinger Equation in the Plane

A Dissection of Bukhgeim's Result

Forskningsoutput: AvhandlingLicentiatavhandling Avhandlingar

Sammanfattning

The purpose of this licentiate thesis is to present Bukhgeim's result of 2007, which solves the inverse boundary value problem of the Schrödinger equation in the plane. The thesis is mainly based on Bukhgeim's paper [B] and Astala's presentation on the matter, 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 Schrödinger 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 [B] with the stationary phase method based on 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 Schrödinger equations with the same boundary data we get an orthogonality relation for the solutions of the different 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 Schrödinger equations are the same.

In the last section we contemplate an unclear detail in [B]. It seems that without more details Bukhgeim's result shows the solvability of the inverse problem only for differentiable potentials instead of ones in Lp(Ω). There is also a brief summary of Astala's presentation and a remark about an unclarity which surfaced during the writing of this thesis.

Originalspråkengelska
StatusPublicerad - jun 2010
MoE-publikationstypG3 Licentiatavhandling

Vetenskapsgrenar

  • 111 Matematik

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title = "The Inverse Problem of the Schr{\"o}dinger Equation in the Plane: A Dissection of Bukhgeim's Result",
abstract = "The purpose of this licentiate thesis is to present Bukhgeim's result of 2007, which solves the inverse boundary value problem of the Schr{\"o}dinger equation in the plane. The thesis is mainly based on Bukhgeim's paper [B] and Astala's presentation on the matter, 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{\"o}dinger 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 [B] with the stationary phase method based on 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{\"o}dinger equations with the same boundary data we get an orthogonality relation for the solutions of the different 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{\"o}dinger equations are the same. In the last section we contemplate an unclear detail in [B]. It seems that without more details Bukhgeim's result shows the solvability of the inverse problem only for differentiable potentials instead of ones in Lp(Ω). There is also a brief summary of Astala's presentation and a remark about an unclarity which surfaced during the writing of this thesis.",
keywords = "111 Mathematics, inverse problem, Schr{\"o}dinger equation, stationary phase",
author = "Eemeli Bl{\aa}sten",
year = "2010",
month = "6",
language = "English",

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The Inverse Problem of the Schrödinger Equation in the Plane : A Dissection of Bukhgeim's Result. / Blåsten, Eemeli.

2010. 40 s.

Forskningsoutput: AvhandlingLicentiatavhandling Avhandlingar

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T2 - A Dissection of Bukhgeim's Result

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N2 - The purpose of this licentiate thesis is to present Bukhgeim's result of 2007, which solves the inverse boundary value problem of the Schrödinger equation in the plane. The thesis is mainly based on Bukhgeim's paper [B] and Astala's presentation on the matter, 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 Schrödinger 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 [B] with the stationary phase method based on 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 Schrödinger equations with the same boundary data we get an orthogonality relation for the solutions of the different 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 Schrödinger equations are the same. In the last section we contemplate an unclear detail in [B]. It seems that without more details Bukhgeim's result shows the solvability of the inverse problem only for differentiable potentials instead of ones in Lp(Ω). There is also a brief summary of Astala's presentation and a remark about an unclarity which surfaced during the writing of this thesis.

AB - The purpose of this licentiate thesis is to present Bukhgeim's result of 2007, which solves the inverse boundary value problem of the Schrödinger equation in the plane. The thesis is mainly based on Bukhgeim's paper [B] and Astala's presentation on the matter, 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 Schrödinger 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 [B] with the stationary phase method based on 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 Schrödinger equations with the same boundary data we get an orthogonality relation for the solutions of the different 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 Schrödinger equations are the same. In the last section we contemplate an unclear detail in [B]. It seems that without more details Bukhgeim's result shows the solvability of the inverse problem only for differentiable potentials instead of ones in Lp(Ω). There is also a brief summary of Astala's presentation and a remark about an unclarity which surfaced during the writing of this thesis.

KW - 111 Mathematics

KW - inverse problem

KW - Schrödinger equation

KW - stationary phase

M3 - Licenciate's thesis

ER -