Strong scarring of logarithmic quasimodes

Eswarathasan, Suresh; Nonnenmacher, Stéphane

doi:10.5802/aif.3137

Strong scarring of logarithmic quasimodes
[Quasimode logarithmique et grosse balafre]

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)

Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2307-2347.

Résumé
Abstract

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 $ε ℏ / | log ℏ |$ , 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.

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 $ε ℏ / | log ℏ |$ , 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.

Reçu le : 2015-08-17
Accepté le : 2017-02-06
Publié le : 2017-12-14

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

Affiliations des auteurs :

Eswarathasan, Suresh ¹ ; Nonnenmacher, Stéphane ^{2, 3}

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Eswarathasan, Suresh; Nonnenmacher, Stéphane. Strong scarring of logarithmic quasimodes. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2307-2347. doi : 10.5802/aif.3137. http://archive.numdam.org/articles/10.5802/aif.3137/

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