Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms

TL;DR

This paper demonstrates that coherent states under perturbed hyperbolic automorphisms on the two-torus become uniformly distributed over time, and it provides insights into the extent of eigenstate localization or scarring.

Contribution

It introduces a logarithmic time-scale Egorov theorem for quantized perturbed hyperbolic automorphisms, linking quantum dynamics with classical equidistribution and scarring control.

Findings

Coherent states equidistribute on the torus over logarithmic time scales.

The method controls the potential strong scarring of eigenstates.

Provides a new tool for analyzing quantum chaos on the torus.

Abstract

We show that on a suitable time scale, logarithmic in $\hbar$, the coherent states on the two-torus, evolved under a quantized perturbed hyperbolic toral automorphism, equidistribute on the torus. We then use this result to obtain control on the possible strong scarring of eigenstates of the perturbed automorphisms by periodic orbits. Our main tool is an adapted Egorov theorem, valid for logarithmically long times.