Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures.

*(English)*Zbl 1319.35207Summary: We look at the long-time behavior of solutions to a semi-classical Schrödinger equation on the torus. We consider time scales which go to infinity when the semi-classical parameter goes to zero and we associate with each time-scale the set of semi-classical measures associated with all possible choices of initial data. On each classical invariant torus, the structure of semi-classical measures is described in terms of two-microlocal measures, obeying explicit propagation laws. We apply this construction in two directions. We first analyze the regularity of semi-classical measures, and we emphasize the existence of a threshold: for time-scales below this threshold, the set of semi-classical measures contains measures which are singular with respect to Lebesgue measure in the “position” variable, while at (and beyond) the threshold, all the semi-classical measures are absolutely continuous in the “position” variable, reflecting the dispersive properties of the equation. Second, the techniques of two-microlocal analysis introduced in the paper are used to prove semiclassical observability estimates. The results apply as well to general quantum completely integrable systems.

##### MSC:

35Q41 | Time-dependent Schrödinger equations and Dirac equations |

35R06 | PDEs with measure |

35B65 | Smoothness and regularity of solutions to PDEs |

35P15 | Estimates of eigenvalues in context of PDEs |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

93B07 | Observability |

81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |