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Résonances du laplacien sur les variétés à pointes

Abstract : In this thesis, we study the resonances of the Laplace operator on cusp manifolds. They are manifolds whose ends are real hyperbolic cusps. The resonances were introduced by Selberg in the 50's for the constant curvature cusp surfaces. Their definition was later extended to the case of variable curvature by Lax and Phillips. The resonances are the poles of a meromorphic family of generalized eigenfunctions of the Laplace operator. They are associated to the continuous spectrum of the Laplace operator. To analyze this continuous spectrum, different directions of research are investigated.On the one hand, we obtain results on the localization of resonances. In particular, if the curvature is negative, for a generic set of metrics, they split into two sets. The first one is included in a band near the spectrum. The other is composed of resonances that are far from the spectrum. This leaves a log zone without resonances. On the other hand, we study the microlocal measures associated to certain sequences of spectral parameters. In particular we show that for some sequences of parameters that converge to the spectrum, but not too fast, the associated microlocal measure has to be the Liouville measure. This property holds when the curvature is negative.
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Submitted on : Monday, September 7, 2015 - 2:47:25 PM
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  • HAL Id : tel-01194752, version 1

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Yannick Bonthonneau. Résonances du laplacien sur les variétés à pointes. Mathématiques générales [math.GM]. Université Paris Sud - Paris XI, 2015. Français. ⟨NNT : 2015PA112141⟩. ⟨tel-01194752⟩

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