Strong Scarring of Logarithmic Quasimodes

Cicatrices de quasimodes logarithmiques

Suresh Eswarathasan (1) , Stéphane Nonnenmacher (2, 3)

1 IHES - Institut des Hautes Études Scientifiques
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider a semiclassical (pseudo)differential operator on a compact surface ($M, g$), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit $\gamma$ at some energy $E_0$. For any $\varepsilon$ $>$0, we then explicitly construct families of $quasimodes$ of this operator, satisfying an energy width of order $\varepsilon$$\hbar\over{|log \hbar|}$ in the semiclassical limit, but which still exhibit a " strong scar " on the orbit $\gamma$, i.e. that these states have a positive weight in any microlocal neighbourhood of $\gamma$. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks [Br13] in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.

Mots clés: quantum normal form; gaussian beam; semiclassical analysis; quantum normal form

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Suresh Eswarathasan, Stéphane Nonnenmacher. Strong Scarring of Logarithmic Quasimodes. Annales de l'Institut Fourier, 2017, Annales de l'Institut Fourier, 67 (6), pp.2307-2347. ⟨hal-01255323v2⟩