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Etats propres de systèmes classiquement chaotiques dans l'espace des phases

Abstract : The aim of this work is to study classically chaotic quantum systems. We restrict ourselves to one-dimensional dynamics, and pay a particular attention to eigenstates, using both analytical and numerical methods. Quantum states are represented using phase space probability densities (Husimi densities), so that they can be easily compared to classically invariant measures, in the semiclassical limit. On the other hand, a quantum state can be built directly from the knoowledge of its constellation, i.e. the set of zeros of its Husimi density. We first study an integrable Hamiltonian system with a fixed unstable point. A precise description of Husimi densities of eigenstates near the critical energu is provided by uniform WKB approximations. While densities concentrate exponentially around the separatrix, zeros are stributed along classically defined (anti-Stokes) lines. We then study area-preserving maps on the torus, in particular Arnold's ``cat'' maps and the baker's map, which are both proven to be fully chaotic and for which we know a consistent quantization procedure. Due to arithmetical properties of cat maps, we can build families of very ergodic eigenstates, for which the constellations form crystals on the torus. More generally, we show that eigenstates of these quantum maps have, on average, similar properties to Gaussian random states: their Husimi densities and constellations are grossly equidistributed over the whole torus in the semiclassical limit, and their fluctuations around the ergodic measure are universal. On the other hand, we argue that the specic features of an individual eigenstate (e.g. a scar above a periodic point) can be robustly extracted from the first few Fourier coefficients of its constellation.
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Submitted on : Thursday, November 8, 2001 - 4:43:11 PM
Last modification on : Wednesday, September 12, 2018 - 2:13:57 PM
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  • HAL Id : tel-00000855, version 1



Stéphane Nonnenmacher. Etats propres de systèmes classiquement chaotiques dans l'espace des phases. Physique mathématique [math-ph]. Université Paris Sud - Paris XI, 1998. Français. ⟨tel-00000855⟩



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