Règles de quantification semi-classique pour une orbite périodique de type hyberbolique

Abstract : In this Thesis we consider semi-excited resonances for a h-Pseudo-Differential Operator (h-PDO for short) H(x, hDx; h) on L2(M) induced by a periodic orbit of hyperbolic type at energy E = 0, as arises when M = Rn and H(x, hDx; h) is Schrödinger operator withAC Stark effect, or H(x, hDx; h) is the geodesic flow on an axially symmetric manifold M,extending Poincaré example of Lagrangian systems with 2 degree of freedom. We generalizethe framework of Gérard and Sjöstrand, in the sense that we allow for hyperbolic and ellipticeigenvalues of Poincaré map, and look for (excited) resonances with imaginary part of magnitude hs, with 0 < s < 1,It is known that these resonances are given by the zeroes of a determinant associatedwith Poincaré map. We make here this result more precise, in providing a first order asymptoticsof Bohr-Sommerfeld quantization rule in terms of the (real) longitudinal and (complex)transverse quantum numbers, including the action integral, the sub-principal 1-form and Gelfand-Lidskii index.
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Hanen Louati. Règles de quantification semi-classique pour une orbite périodique de type hyberbolique. Mathématiques générales [math.GM]. Université de Toulon; Université de Tunis El-Manar. Faculté des Sciences de Tunis (Tunisie), 2017. Français. ⟨NNT : 2017TOUL0004⟩. ⟨tel-01684763⟩

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