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Asymptotic solutions and resonances for Klein-Gordon and Schrödinger operators

Abstract : My PhD thesis deals with semi-classical analysis. It is divided in three parts. In the first one, we have considered a semi-classical Klein-Gordon operator in the one dimensional case. WKB constructions exist in the region where the potential relies below the energy level. Under some hypothesis, we have proved that these solutions can be extended beyond this region, thanks to methods used near turning points for the Schrödinger operator. We have then studied an example where explicit calculations can be made. Lastly, we have obtained new estimates for eigenfunctions in any dimension, if the gradient of the Agmon distance is lipschitzian. The second part of this work concerns resonances for the Schrödinger operator in the one dimensional case. When the potential presents two compact wells and a infinite one, for energy level under consideration, we have obtained conditions for anti-crossing of resonances as well as there graphics. This can be done constructing modes for the operator. For any given number of compact wells, it also led to estimates for imaginary part of resonances, when an simple interaction occurs. Finally, in the third part of this work, we have considered a Schrödinger operator which potential presents a non degenerate maximum. We have studied resonances generated by an homocline curve which contains this maximum. In the one dimansional case, a quantification equation which led to the resonances we look for is obtained. In the n-dimensional case, an out-going asymptotic solution along the curve is constructed, by adapting B. Helffer and J. Sjöstrand's method for the bottom of non resonant wells. An FBI transform allows then to guess a first level of résonances.
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https://tel.archives-ouvertes.fr/tel-00002342
Contributor : Emmanuelle Amar-Servat <>
Submitted on : Friday, January 31, 2003 - 10:52:52 AM
Last modification on : Saturday, February 15, 2020 - 1:41:59 AM
Long-term archiving on: : Friday, April 2, 2010 - 6:23:01 PM

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  • HAL Id : tel-00002342, version 1

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Emmanuelle Amar-Servat. Asymptotic solutions and resonances for Klein-Gordon and Schrödinger operators. Mathématiques [math]. Université Paris-Nord - Paris XIII, 2002. Français. ⟨tel-00002342⟩

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