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 arbitrairement petit, nous construisons une famille de quasimodes de cet opérateur, dont la largeur en énergie est d’ordre , 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 , we construct semiclassical families of quasimodes of this operator, with energy widths of order , 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.
Accepté le :
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Classification : 35-xx, 58Jxx, 37-xx
Mots clés : analyse semiclassique, quasimode, unique ergodicité quantique, balafre d’orbite périodique
@article{AIF_2017__67_6_2307_0, author = {Eswarathasan, Suresh and Nonnenmacher, St\'ephane}, title = {Strong scarring of logarithmic quasimodes}, journal = {Annales de l'Institut Fourier}, pages = {2307--2347}, publisher = {Association des Annales de l'institut Fourier}, volume = {67}, number = {6}, year = {2017}, doi = {10.5802/aif.3137}, language = {en}, url = {https://aif.centre-mersenne.org/item/AIF_2017__67_6_2307_0/} }
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. https://aif.centre-mersenne.org/item/AIF_2017__67_6_2307_0/
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