Quantisations of piecewise affine maps on the torus and their quantum limits

Cheng-Hung Chang (1) , Tyll Krueger (2) , Roman Schubert (3) , Serge Troubetzkoy (4, 5, 6)

1 NCTU Institute of Physics
2 Institut für Mathematik [Berlin]
3 School of Mathematics [Bristol]
4 IML - Institut de mathématiques de Luminy
5 CPT - Centre de Physique Théorique - UMR 6207
6 CPT - Centre de Physique Théorique - UMR 7332

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking.

Domaines: Systèmes dynamiques [math.DS]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]

Cheng-Hung Chang, Tyll Krueger, Roman Schubert, Serge Troubetzkoy. Quantisations of piecewise affine maps on the torus and their quantum limits. Communications in Mathematical Physics, 2008, 282 (2), pp.395-418. ⟨hal-00142663⟩