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The full delocalization of eigenstates for the quantized cat map

Nir Schwartz

TL;DR

The paper addresses the delocalization problem for eigenstates of the quantum cat map on the torus, proving that every semiclassical measure is fully supported on 𝕋². The authors develop an anisotropic semiclassical calculus adapted to toral hyperbolic dynamics, propagate refined partitions of unity along the quantum evolution, and apply a fractal uncertainty principle to bound the uncontrolled region. They derive a quantitative main estimate linking the state norm to its localization under the quantum map and a spectral error term, yielding lower bounds on mass in any open set and hence full delocalization in the semiclassical limit. This work extends fractal-uncertainty-based delocalization results from continuous-time hyperbolic manifolds to discrete toral dynamics, with potential generalizations to higher-dimensional cat maps and Sp(2d,ℤ).

Abstract

We consider the quantum cat map - a toy model of a quantized chaotic system. We show that its eigenstates are fully delocalized on $\mathbb{T}^2$ in the semiclassical limit (or equivalently that each semiclassical measure is fully supported on $\mathbb{T}^2$). We adapt the proof of a similar result proved for the eigenstates of $-Δ_g$ on compact hyperbolic surfaces from [arXiv:1705.05019], relying on the fractal uncertainty principle in [arXiv:1612.09040].

The full delocalization of eigenstates for the quantized cat map

TL;DR

The paper addresses the delocalization problem for eigenstates of the quantum cat map on the torus, proving that every semiclassical measure is fully supported on 𝕋². The authors develop an anisotropic semiclassical calculus adapted to toral hyperbolic dynamics, propagate refined partitions of unity along the quantum evolution, and apply a fractal uncertainty principle to bound the uncontrolled region. They derive a quantitative main estimate linking the state norm to its localization under the quantum map and a spectral error term, yielding lower bounds on mass in any open set and hence full delocalization in the semiclassical limit. This work extends fractal-uncertainty-based delocalization results from continuous-time hyperbolic manifolds to discrete toral dynamics, with potential generalizations to higher-dimensional cat maps and Sp(2d,ℤ).

Abstract

We consider the quantum cat map - a toy model of a quantized chaotic system. We show that its eigenstates are fully delocalized on in the semiclassical limit (or equivalently that each semiclassical measure is fully supported on ). We adapt the proof of a similar result proved for the eigenstates of on compact hyperbolic surfaces from [arXiv:1705.05019], relying on the fractal uncertainty principle in [arXiv:1612.09040].

Paper Structure

This paper contains 17 sections, 33 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quantum dynamics on the torus
  3. Hyperbolic automorphisms
  4. Anisotropic calculi associated to Lagrangian foliations
  5. The quantum settings
  6. Example: Degli—Esposti's cat map
  7. The Quantum space of the torus phase space
  8. The main estimate
  9. Geometric construction and propagated operators
  10. Outline of the proof
  11. An estimate on the controlled region $\mathcal{Y}$
  12. $\nu$-porous sets used for proving \ref{['propfup']}
  13. Introducing a partition of unity by smooth cut-offs on $\mathbb{R}^2$
  14. An estimate on words in the uncontrolled region
  15. Proof of \ref{['propfup']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\gamma\in \tilde{\Gamma}(2)$ (where $\tilde{\Gamma}(2)$ is defined in Gamma2) be a hyperbolic matrix quantized into the family $\{\mathcal{M}_{N}(\gamma)\}_N$. Let $\mu_{\text{sc}}$ be an associated semiclassical measure. Then for every open $\emptyset\neq\Omega\subset\mathbb{T}^2$ there exists

Figures (2)

  • Figure 1: Applying a hyperbolic toral automorphism $\gamma$ to a cluster of points (in orange) stretches them along the unstable branches, eventually filling densely the torus.
  • Figure 2: Given a symbol $a\in C^\infty(\mathbb{T}^2)$ we construct in \ref{['gcon']} a partition of unity on $\mathbb{T}^2\cong \left[0,1\right)^2$. $\mathbb{R}^2$ is equipped with $(y,\eta)$ coordinates. The $(x,\xi)$ coordinates are obtained after applying the coordinate map $\iota$ to the coordinates $(y,\eta)$. The new coordinates describe the decomposition of $\mathbb{R}^2$ into stable and unstable directions. The yellow domain represents $\text{supp}\ (a_1)$ and the domain in green lines represents $\text{supp}\ (a_2)$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2: The "delocalization" of the eigenfunctions
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Lemma 3
  • proof
  • Remark 1
  • ...and 61 more