Upper bounds on the rate of quantum ergodicity

TL;DR

This paper investigates the speed at which quantum eigenfunctions become uniformly distributed in systems with ergodic classical limits, providing upper bounds on the rate of quantum ergodicity and extending results to various quantum maps.

Contribution

It derives explicit logarithmic bounds on the rate of quantum ergodicity for ergodic and weakly mixing systems, generalizing previous work and applying to quantized maps on the torus.

Findings

Upper bound of order |ln(ħ)|^{-1} on quantum ergodicity rate.

Logarithmic rate for perturbed cat-maps.

Algebraic rate for parabolic maps.

Abstract

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.