Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms

TL;DR

This paper demonstrates that for certain perturbed hyperbolic automorphisms of the torus, some eigenstates localize on invariant submanifolds despite the classical ergodic behavior, revealing scarring phenomena.

Contribution

It introduces the existence of eigenstates that localize on invariant submanifolds in perturbed quantum automorphisms, showing a form of scarring in a new class of systems.

Findings

Existence of eigenstates localized on invariant submanifolds.

Almost all eigenstates converge to the volume measure of the torus.

Some eigenstates concentrate on specific invariant submanifolds.

Abstract

We exhibit scarring for certain nonlinear ergodic toral automorphisms. There are perturbed quantized hyperbolic toral automorphisms preserving certain co-isotropic submanifolds. The classical dynamics is ergodic, hence in the semiclassical limit almost all eigenstates converge to the volume measure of the torus. Nevertheless, we show that for each of the invariant submanifolds, there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold.