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

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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

  • all metrics

  • well defined

  • double well

  • lebesgue measure

  • then

  • riemannian surface

  • sequence

  • banach space

  • ty ?

  • choosing then


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Modes and quasi-modes on surfaces: variation on an idea of Andrew Hassell
YvesColindeVerdi`ere
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. Wewill show that, foralmost allRiemannian metrics on closed surfaces with an elliptic generic closed geodesicγ, there exists exact modes located onγ. Similar problems in the integrable case are discussed in several papers of J. Toth and S. Zelditch (see [8]).
2 Quasi-modesassociated to an elliptic generic closed geodesic 2.1 Babich-Lazutkinand Ralston quasi-modes Definition 2.1A periodic geodesicγon a Riemannian surface(X, g)is said to be elliptic genericopame´racnioPdezfneaveegifihtearielinofthluesγare of modulus 1and are not roots of the unity. Theorem 2.1 (Babich-Lazutkin [2], Ralston [6, 7])Ifγis an elliptic generic closed geodesic of periodT >0on a closed Riemannian surface(X, g), there exists 2 a sequence ofquasi-modes (um)mNofL(X, dxg)norm equal to1which satisfies −∞ • kgλm)umkL(X,dxg)=O(m) 2 1 There existsαso that  2 2πm+α λm= +O(1) T R 2−∞ For any compactKdisjoint ofγ,|um|=O(m). K Corollary 2.1There exists a sub-sequence(µjm)mNof the spectrum(µj)jNof −∞ the Laplace operator so thatµjm=λm+O(m). GrenobleUniversity,InstitutFourier,Unit´emixtederechercheCNRS-UJF5582,BP74, 38402-SaintMartindH`eresCedex(France);yves.colin-de-verdiere@ujf-grenoble.fr 1 1 αis given byα= (m1+ )θ+wherem1Nis a “transverse” quantum number, the 2 linearizedPoincare´mapisarotationofangleθ(0< θ <2π) andp= 0 or 1 is a “Maslov index” ofγ
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