Long time semiclassical Egorov theorem for ℏ-pseudodifferential systems

Article type: Research Article

Authors: Assal, Marouane

Affiliations: Institut de Mathématiques de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France. E-mail: marouane.assal@math.u-bordeaux1.fr

Abstract: In the Heisenberg picture, we study the semiclassical time evolution of a bounded quantum observable Qw(x,ℏDx;ℏ) associated to a (m×m) matrix-valued symbol Q generated by a semiclassical matrix-valued Hamiltonian H∼H0+ℏH1. Under a non-crossing assumption on the eigenvalues of the principal symbol H0 that ensures the existence of almost invariant subspaces of L2(Rn)⊗Cm, and for a class of observables that are semiclassically block-diagonal with respect to the projections onto these almost invariants subspaces, we establish a long time matrix-valued version for the semiclassical Egorov theorem valid in a large time interval of Ehrenfest type T(ℏ)≃log(ℏ−1).

Keywords: Egorov theorem, quantum evolution, semiclassical approximation, ℏ-pseudodifferential systems

DOI: 10.3233/ASY-161394

Journal: Asymptotic Analysis, vol. 101, no. 1-2, pp. 17-67, 2017