# Contribution to the study of the Schrödinger equation : inverse problem in a bounded domain and bilinear optimal control of an Hartree-Fock equation.

Abstract : This thesis deals with some properties of the time dependent Schrödinger equation. On the one hand, we study an inverse problem about this equation set in a bounded domain, with time independent potential and Dirichlet boundary data. Using a Carleman estimate, we prove the well-posedness of the inverse problem of determining the potential from measurements of the normal derivative of the solution through a part of the boundary. On the other hand, we consider a Schrödinger equation set in $\mathbb R^3$ with a coulombian potential, locally singular, and an electric unbounded potential, both depending on space and time variables. We prove the existence of a unique solution, as regular as the initial data, for the linear equation and for an equation with Hartree nonlinearity. This is a first step for the study of a system where this Hartree-Fock equation is coupled with classical newtonian dynamics. Eventually, we consider bilinear optimal control problems of the solution of these different equations, the control being performed by the external electric potential. We prove the existence of optimal controls and give optimality conditions in the suitable cases.
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Cited literature [60 references]

https://tel.archives-ouvertes.fr/tel-00007684
Contributor : Lucie Baudouin <>
Submitted on : Wednesday, December 8, 2004 - 5:16:34 PM
Last modification on : Tuesday, November 17, 2020 - 1:50:03 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:25:24 PM

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• HAL Id : tel-00007684, version 1

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Lucie Baudouin. Contribution to the study of the Schrödinger equation : inverse problem in a bounded domain and bilinear optimal control of an Hartree-Fock equation.. Mathematics [math]. Université de Versailles-Saint Quentin en Yvelines, 2004. English. ⟨tel-00007684⟩

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