Long time semiclassical Egorov theorem for h-pseudodifferential systems

What is it about?

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).

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