000002024 001__ 2024
000002024 005__ 20240523045555.0
000002024 0247_ $$2doi$$a10.6082/uchicago.2024
000002024 041__ $$aen
000002024 245__ $$aReconstruction of the Magnetic Field for a Schrödinger Operator in a Cylindrical Setting
000002024 260__ $$bThe University of Chicago
000002024 269__ $$a2019-08
000002024 300__ $$a125
000002024 336__ $$aDissertation
000002024 502__ $$bPh.D.
000002024 520__ $$aIn this thesis we consider a magnetic Schrödinger inverse problem over a compact domain contained in an infinite cylindrical manifold. We show that, under certain conditions on the electromagnetic potentials, we can recover the magnetic field from boundary measurements in a constructive way. A fundamental tool for this procedure is a global Carleman estimate for the magnetic Schrödinger operator. We prove this by conjugating the magnetic operator essentially into the Laplacian, and using the Carleman estimates for it proven by Kenig--Salo--Uhlmann in the anisotropic setting, see [KSU11a]. The conjugation is achieved through pseudodifferential operators over the cylinder, for which we develop the necessary results.

The main motivations to attempt this question are the following results concerning the magnetic Schrödinger operator: first, the solution to the uniqueness problem in the cylindrical setting in [DSFKSU09], and, second, the reconstruction algorithm in the Euclidean setting from [Sal06]. We will also borrow ideas from the reconstruction of the electric potential in the cylindrical setting from [KSU11b]. These two new results answer partially the Carleman estimate problem (Question 4.3.) proposed in [Sal13] and the reconstruction for the magnetic Schrödinger operator mentioned in the introduction of [KSU11b]. To our knowledge, these are the first global Carleman estimates and reconstruction procedure for the magnetic Schrödinger operator available in the cylindrical setting.
000002024 542__ $$fUniversity of Chicago dissertations are covered by copyright.
000002024 650__ $$aMathematics
000002024 653__ $$aDirichlet-to-Neumann map
000002024 653__ $$aInverse problem
000002024 653__ $$aMagnetic Schrödinger operator
000002024 653__ $$aReconstruction
000002024 690__ $$aPhysical Sciences Division
000002024 691__ $$aMathematics
000002024 7001_ $$aCampos Salas, Daniel$$uUniversity of Chicago
000002024 72012 $$aCarlos E. Kenig
000002024 72014 $$aGuillaume Bal
000002024 8564_ $$9922cbe14-dff3-4926-ae2e-dc2fbb152171$$ePublic$$s579607$$uhttps://knowledge.uchicago.edu/record/2024/files/CamposSalas_uchicago_0330D_14996.pdf
000002024 909CO $$ooai:uchicago.tind.io:2024$$pDissertations$$pGLOBAL_SET
000002024 983__ $$aDissertation