Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

J. M. Bouclet, S. De Bievre

TL;DR

The paper develops a framework to control quantum dynamics of perturbed hyperbolic toral maps at times up to a logarithmic scale in the semiclassical parameter. It proves an explicit Egorov-type expansion for perturbed toral automorphisms and combines it with classical exponential mixing to show that coherent states become equidistributed on the torus within a logarithmic time window. This enables partial control of strong scarring by periodic orbits and yields constraints on the semiclassical limits of eigenfunctions. The methods provide a dynamical, non-arithmetic route to quantum ergodicity-type results on the torus with perturbations and link quantum evolution to classical invariant measures through precise remainder estimates.

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.

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

TL;DR

The paper develops a framework to control quantum dynamics of perturbed hyperbolic toral maps at times up to a logarithmic scale in the semiclassical parameter. It proves an explicit Egorov-type expansion for perturbed toral automorphisms and combines it with classical exponential mixing to show that coherent states become equidistributed on the torus within a logarithmic time window. This enables partial control of strong scarring by periodic orbits and yields constraints on the semiclassical limits of eigenfunctions. The methods provide a dynamical, non-arithmetic route to quantum ergodicity-type results on the torus with perturbations and link quantum evolution to classical invariant measures through precise remainder estimates.

Abstract

We show that on a suitable time scale, logarithmic in , 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.

Paper Structure

This paper contains 8 sections, 19 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Weyl quantization and Egorov Theorem
  3. Perturbations of quantized hyperbolic maps
  4. Equirepartition of time-evolved localized states
  5. The example of (generalized) coherent states
  6. Semiclassical behavior of eigenstates
  7. A mixing theorem for perturbations of hyperbolic maps on $\mathbb T^{2}$.
  8. Generalized coherent states decompositions

Key Result

Proposition 2.1

There exists a bilinear map $(f,g) \mapsto f \# g$ from ${\mathcal{B}}({\mathcal{M}})^2$ to ${\mathcal{B}}({\mathcal{M} })$ such that The function $f \# g$ has a full asymptotic expansion in powers of $\hbar$, meaning that for all integers $J$ where $f \#_j g = \sum_{|\alpha + \beta| = j} \Gamma(\alpha,\beta) \partial_q^{\alpha} \partial_p^{\beta} f \partial_q^{\beta}

Theorems & Definitions (19)

  • Proposition 2.1
  • Theorem 2.2: Egorov theorem
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 9 more