Une étude de diffusion inverse pour l'équation de Schrödinger avec champ électromagnétique

Abstract : This thesis studies direct and inverse scattering problems for the Scrodinger equation with electric and magnetic fields. We consider both, stationary and time-dependent methods. First, we give a brief description of the physical aspects of the Aharonov-Bohm effect. Then, we give rigorous descrition of the formula for the propagator known in the physics literature as "Feynman's rule". This formula shows that in the semiclassicl regim the propagator is given by the exponential of a phase times the unperturbed propagator. The phase is the circulation of the magnetic potential. In the next chapter, we study a scattering process in an electromagnetic field and we obtain a representation formula for the scattering matrix. Then, the inverse problem in the case of electromagnetic fields is studied with a new stationary method. This approach gives a complete asymptotic expansion at high energies of the scattering operator between appropriate chosen states. The Aharonov-Bohm effect is considered (a scattering problem in exterior domains). It is proven that in two dimensions, the flux of the magnetic field is given modulo by the scattering operator. Finally, the inverse scattering problem when the given scattering data is given in an energy interval is studied. it is proven that the given scattering data determines the asyptotic of the potential at infinity. We also study Stark effect and Schrödinger operators with repulsive potentials with the Enss-Weder time dependent method.
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Mathématiques [math]. Université de Nantes, 2004
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Contributor : François Nicoleau <>
Submitted on : Monday, January 3, 2005 - 3:50:20 PM
Last modification on : Friday, March 27, 2015 - 10:00:12 AM
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  • HAL Id : tel-00007889, version 1



François Nicoleau. Une étude de diffusion inverse pour l'équation de Schrödinger avec champ électromagnétique. Mathématiques [math]. Université de Nantes, 2004. <tel-00007889>



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