no. 6
Strong scarring of logarithmic quasimodes
Eswarathasan, Suresh1; Nonnenmacher, Stéphane2, 3
1 Department of Mathematics and Statistics McGill University 805 Rue Sherbrooke Ouest Montréal (Canada)
2 Institut de Physique Théorique Université Paris-Saclay Commissariat à l’énergie atomique 91191 Gif-sur-Yvette (France)
3 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay (France)
Annales de l'Institut Fourier, Volume 67 (2017) no. 6, pp. 2307-2347.

We consider a semiclassical pseudodifferential operator on a compact surface, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit at some energy. For an arbitrary small ε>0, we construct semiclassical families of quasimodes of this operator, with energy widths of order εℏ/|logℏ|, and which feature a strong scar along that hyperbolic orbit. Our construction proceeds by controlling the evolution of Gaussian wavepackets up to the Ehrenfest time.

Nous considérons un opérateur pseudodifférentiel semiclassique sur une surface compacte, tel que le flot Hamiltonien engendré par son symbole principal possède, à une certaine énergie, une orbite périodique hyperbolique. Pour un paramètre ε>0 arbitrairement petit, nous construisons une famille de quasimodes de cet opérateur, dont la largeur en énergie est d’ordre εℏ/|logℏ|, mais qui possèdent un poids positif (une «  grosse balafre ») autour de cette orbite périodique. Notre construction procède par un contrôle de l’évolution de paquets d’onde gaussiens jusqu’au temps d’Ehrenfest.

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Accepted:
Published online:
DOI: 10.5802/aif.3137
Classification: 35-xx, 58Jxx, 37-xx
Keywords: semiclassical analysis, quasimode, QUE, strong scarring
Mot clés : analyse semiclassique, quasimode, unique ergodicité quantique, balafre d’orbite périodique
Eswarathasan, Suresh 1; Nonnenmacher, Stéphane 2, 3

1 Department of Mathematics and Statistics McGill University 805 Rue Sherbrooke Ouest Montréal (Canada)
2 Institut de Physique Théorique Université Paris-Saclay Commissariat à l’énergie atomique 91191 Gif-sur-Yvette (France)
3 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Eswarathasan, Suresh; Nonnenmacher, Stéphane. Strong scarring of logarithmic quasimodes. Annales de l'Institut Fourier, Volume 67 (2017) no. 6, pp. 2307-2347. doi : 10.5802/aif.3137. https://aif.centre-mersenne.org/articles/10.5802/aif.3137/

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