Upper bounds on the rate of quantum ergodicity

TL;DR

The paper analyzes the semiclassical behavior of eigenfunctions for quantum systems with ergodic classical limits, where the quantum ergodicity theorem implies equidistribution in a weak sense. It presents a simple derivation giving an upper bound of order $|\ln \hbar|^{-1}$ on the rate of quantum ergodicity under a rate assumption on the classical ergodicity, and a analogous bound on transition amplitudes when the classical system is weakly mixing. The results generalize Zelditch's earlier work and are extended to certain quantised maps on the torus, yielding a logarithmic rate for perturbed cat maps and a sharp algebraic rate for parabolic maps. These bounds clarify how fast quantum states spread in phase space as the semiclassical parameter $\hbar \to 0$, linking classical mixing properties to quantum distribution rates.

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.