Upper bounds on the rate of quantum ergodicity

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.

18 pages

Cited by in corpus (9)

• Eigenstate Thermalization Hypothesis for Wigner Matrices
• Eigenfunction Statistics on Quantum Graphs
• Small scale quantum ergodicity in negatively curved manifolds
• Fluctuations in local quantum unique ergodicity for generalized Wigner matrices
• Small scale equidistribution of eigenfunctions on the torus
• Some open questions in "wave chaos"
• Quantum ergodicity for quantum graphs without back-scattering
• Rényi and Tsallis entropies related to eigenfunctions of quantum graphs
• Log-scale equidistribution of zeros of quantum ergodic eigensections