Mathematics

The problem of existence and direct construction of adiabatic invariants - i.e. approximate first integrals - of perturbed Hamiltonian systems is relevant to the foundations of quantum (as well as classical) statistical mechanics. Thermal relaxation to the statistical equilibrium is prevented, or essentially modified by the presence of exact constants of motion, and slowed down by the adiabatic invariants, which is the mechanism producing quasi (equilibrium) states described by the generalized (i.e. reduced) Gibbs ensembles, a key ingredient in the so-called pre-thermalization scenario. A scheme for the direct construction of quantum adiabatic invariants is presented. In such a framework, a quantum formulation of the Poincare' theorem of nonexistence of nontrivial invariants under generic perturbations is obtained. As a consequence, one gets the necessary conditions under which the formal construction of an adiabatic invariant of given perturbative order is possible. Moreover, an estimate of the thermalization time of the given invariant, in terms of quantum statistical expectations, is also provided, showing that the higher the order of the invariant, the longer is the relaxation time, as expected. All the results obtained in the quantum framework are based on the algebraic properties of Hamiltonian systems and, as a byproduct, one gets a coordinate-free reformulation of the known classical results.