Modes and quasi modes on surfaces: variation on an idea of Andrew Hassell

Niveau: Supérieur, Licence, Bac+2
Modes and quasi-modes on surfaces: variation on an idea of Andrew Hassell Yves Colin de Verdiere? February 10, 2009 1 Introduction This paper is inspired from the nice idea of A. Hassell in [5]. From the classical paper of V. Arnol'd [1], we know that quasi-modes are not always close to exact modes. We will show that, for almost all Riemannian metrics on closed surfaces with an elliptic generic closed geodesic ?, there exists exact modes located on ?. Similar problems in the integrable case are discussed in several papers of J. Toth and S. Zelditch (see [8]). 2 Quasi-modes associated to an elliptic generic closed geodesic 2.1 Babich-Lazutkin and Ralston quasi-modes Definition 2.1 A periodic geodesic ? on a Riemannian surface (X, g) is said to be elliptic generic if the eigenvalues of the linearized Poincare map of ? are of modulus 1 and are not roots of the unity. Theorem 2.1 (Babich-Lazutkin [2], Ralston [6, 7]) If ? is an elliptic generic closed geodesic of period T > 0 on a closed Riemannian surface (X, g), there exists a sequence of quasi-modes (um)m?N of L2(X, dxg) norm equal to 1 which satisfies • ?(∆g ? ?m)um?L2(X,dxg) = O(m ?∞) • There exists ? so that1 ?m = ( 2pim+ ? T )2

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