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Theses

Problèmes spectraux inverses pour des opérateurs AKNS et de Schrödinger singuliers sur [0,1]

Abstract : Two operators are studied in this thesis: the radial Schrödinger operator, extracted from the non-relativistic quantum mechanic, and the singular AKNS system, adaptation of the radial Dirac equation coming from relativistic quantum mechanic. In the first part, the direct spectral problem is solved for each equation: we obtain eigenvalues, eigenfunctions and their properties about potentials. Limitations caused by the explicit singularity are pointed out: problems risen by Bessel functions appear inside straightforward calculations for asymptotics and estimations. The second part deals with the resolution of these inverse spectral problems. Thanks to the transformation operators, we avoid difficulties created by the singularity. They help us to develop an inverse spectral theory for the singular operators considered. Precisely, we construct a spectral map adapted to study the inverse spectral problem's stability and isospectral sets study. Moreover, a one-to-one result is deduced for singular AKNS and Dirac operators with more regular potentials.
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https://tel.archives-ouvertes.fr/tel-00009719
Contributor : Frédéric Serier <>
Submitted on : Saturday, July 9, 2005 - 5:09:10 PM
Last modification on : Monday, March 25, 2019 - 4:52:05 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:53:09 PM

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  • HAL Id : tel-00009719, version 1

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Frédéric Serier. Problèmes spectraux inverses pour des opérateurs AKNS et de Schrödinger singuliers sur [0,1]. Mathématiques [math]. Université de Nantes, 2005. Français. ⟨tel-00009719⟩

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