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Upper bounds on the rate of quantum ergodicity
We study the semiclassical behaviour of eigenfunctions of quantum systems
with ergodic classical limit. By the quantum ergodicity theorem almost all of
these eigenfunctions become equidistributed in a weak sense. We give a simple
derivation of an upper bound of order \abs{\ln\hbar}^{-1} on the rate of
quantum ergodicity if the classical system is ergodic with a certain rate. In
addition we obtain a similar bound on transition amplitudes if the classical
system is weak mixing. Both results generalise previous ones by Zelditch. We
then extend the results to some classes of quantised maps on the torus and
obtain a logarithmic rate for perturbed cat-maps and a sharp algebraic rate for
parabolic maps.Comment: 18 page