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Quantum dynamical bounds for long-range operators with skew-shift potentials

Wencai Liu, Matthew Powell, Xueyin Wang

Abstract

We employ Weyl's method and Vinogradov's method to analyze skew-shift dynamics on semi-algebraic sets. Consequently, we improve the quantum dynamical upper bounds of Jitomirskaya-Powell, Liu, and Shamis-Sodin for long-range operators with skew-shift potentials.

Quantum dynamical bounds for long-range operators with skew-shift potentials

Abstract

We employ Weyl's method and Vinogradov's method to analyze skew-shift dynamics on semi-algebraic sets. Consequently, we improve the quantum dynamical upper bounds of Jitomirskaya-Powell, Liu, and Shamis-Sodin for long-range operators with skew-shift potentials.

Paper Structure

This paper contains 13 sections, 15 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Exponential sums: Weyl's method
  4. Exponential sums: Vinogradov's method
  5. Diophantine condition
  6. Semi-algebraic set
  7. Fourier analysis
  8. Sublinear bound: Weyl's method
  9. Sublinear bound: Vinogradov's method
  10. Proof of Theorem \ref{['MainTheorem']}
  11. Large deviation, sublinear bound, and quantum dynamics
  12. Proof of Theorem \ref{['MainTheorem']}
  13. Proof of Lemma \ref{['S2b']}

Key Result

Theorem 1.1

Suppose $\omega \in DC(\gamma,\tau)$. Let where $x \in \mathbb{T}^b$, $v$ is real analytic on $\mathbb{T}^b$, and $f$ is the skew-shift on $\mathbb{T}^b.$ Suppose $H_{x,\omega}$ satisfies the LDT (see Section App for the precise definition). Then for any $\phi$ with compact support and $p > 0$ there exists $C=C(\varepsilon, v, A, b, \gamma, where $\delta = \frac{1}{\tau b \psi(b)}$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4: Vinogradov's Mean Value Theorem
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 17 more