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Quantum Unique Ergodicity for maps on the torus

Lior Rosenzweig

TL;DR

This work studies quantum unique ergodicity for maps on the 2-torus, focusing on the Kronecker map and its perturbation. It develops a torus-specific quantization, proves QUE with explicit rate bounds under diophantine conditions, and shows that the perturbed Kronecker map is uniquely ergodic and inherits QUE with a strong rate $O(N^{-2})$ via a conjugacy argument and perturbation quantization. The paper also highlights the nonuniformity of convergence by constructing slow-convergence examples, illustrating the dependence of rates on arithmetic data and chosen rational approximants. These results advance understanding of how classical unique ergodicity translates into quantum ergodicity for toral maps and their perturbations, with implications for rate estimates in quantum chaos on compact phase spaces.

Abstract

When a map is classically uniquely ergodic, it is expected that its quantization will posses quantum unique ergodicity. In this paper we give examples of Quantum Unique Ergodicity for the perturbed Kronecker map, and an upper bound for the rate of convergence.

Quantum Unique Ergodicity for maps on the torus

TL;DR

This work studies quantum unique ergodicity for maps on the 2-torus, focusing on the Kronecker map and its perturbation. It develops a torus-specific quantization, proves QUE with explicit rate bounds under diophantine conditions, and shows that the perturbed Kronecker map is uniquely ergodic and inherits QUE with a strong rate via a conjugacy argument and perturbation quantization. The paper also highlights the nonuniformity of convergence by constructing slow-convergence examples, illustrating the dependence of rates on arithmetic data and chosen rational approximants. These results advance understanding of how classical unique ergodicity translates into quantum ergodicity for toral maps and their perturbations, with implications for rate estimates in quantum chaos on compact phase spaces.

Abstract

When a map is classically uniquely ergodic, it is expected that its quantization will posses quantum unique ergodicity. In this paper we give examples of Quantum Unique Ergodicity for the perturbed Kronecker map, and an upper bound for the rate of convergence.

Paper Structure

This paper contains 14 sections, 19 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Background
  3. QUE for maps on the torus
  4. Background
  5. Notations
  6. Hilbert space of state
  7. Quantum observables
  8. Quantization of maps and rate of convergence
  9. Quantizing Kronecker map
  10. Convergence of eigenstates
  11. Convergence of eigenstates for diophantine pairs:
  12. Proof of theorem \ref{['thm:slow_conv']}
  13. Perturbed Kronecker map
  14. QUE for perturbed Kronecker map

Key Result

Theorem 1.1

Suppose $1,\alpha_1,\alpha_2$ are linearly independent over $\mathbb{Q}$. Let $f\in C^{\infty}(\mathbb{T}^2)$ be a trigonometric polynomial. Then for all eigenfunctions $\psi$ of $U_N(\tau_\alpha)$ we have that for $N$ sufficiently large

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1: "Egorov's Theorem"
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.2
  • ...and 26 more