Publication : t03/070

On the Maximal Scarring for Quantum Cat Map Eigenstates

Faure F. (Laboratoire de Physique et Modélisation des Milieux Condensés (LPM2C), BP 166, 38042 Grenoble Cedex 9, FRANCE)
Nonnenmacher S. (CEA, DSM, SPhT (Service de Physique Théorique), F-91191 Gif-sur-Yvette, FRANCE)

Abstract:
We consider the quantized hyperbolic automorphisms on the $2$-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of ``strong scarring''), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound $1/2$.

Année de publication : 2004
Revue : Commun. Math. Phys. 245 201-214 (2004)
Preprint : arXiv:nlin.CD/0304031
PACS : 81Q50, 81Q20, 81S30, 37D20, 37C40
Keywords : Quantum chaos; Non-unique quantum ergodicity; Strong scarring of eigenstates
Langue : Anglais

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