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Quantum Dynamical Bounds for Quasi-Periodic Operators with Liouville Frequencies

Matthew Bradshaw, Titus de Jong, Wencai Liu, Audrey Wang, Xueyin Wang, Bingheng Yang

TL;DR

The paper addresses quantum dynamics for one-dimensional quasi-periodic Schrödinger operators with Liouville frequencies, establishing sharp upper bounds for the time-averaged spreading of wave packets. It introduces a general transport-criterion based on the existence of a single good Green's function box, and leverages discrepancy bounds for semi-algebraic sets alongside a large deviation theorem tailored to Liouville frequencies. The main contributions are new quantitative upper bounds for Liouville frequencies (and a re-proof of a known bound) and the extension of sublinear-discrepancy techniques to the Liouville setting, providing power–logarithmic and exponential-in-log-log growth bounds for the transport moments. These results broaden the understanding of quantum dynamics beyond Diophantine frequencies and offer a robust framework combining semi-algebraic geometry, Green's function analysis, and transfer-matrix large deviations.

Abstract

We establish quantum dynamical upper bounds for quasi-periodic Schrödinger operators with Liouville frequencies. Our approach combines semi-algebraic discrepancy estimates for the Kronecker sequence $\{nα\}$ with quantitative Green's function estimates adapted to the Liouville setting.

Quantum Dynamical Bounds for Quasi-Periodic Operators with Liouville Frequencies

TL;DR

The paper addresses quantum dynamics for one-dimensional quasi-periodic Schrödinger operators with Liouville frequencies, establishing sharp upper bounds for the time-averaged spreading of wave packets. It introduces a general transport-criterion based on the existence of a single good Green's function box, and leverages discrepancy bounds for semi-algebraic sets alongside a large deviation theorem tailored to Liouville frequencies. The main contributions are new quantitative upper bounds for Liouville frequencies (and a re-proof of a known bound) and the extension of sublinear-discrepancy techniques to the Liouville setting, providing power–logarithmic and exponential-in-log-log growth bounds for the transport moments. These results broaden the understanding of quantum dynamics beyond Diophantine frequencies and offer a robust framework combining semi-algebraic geometry, Green's function analysis, and transfer-matrix large deviations.

Abstract

We establish quantum dynamical upper bounds for quasi-periodic Schrödinger operators with Liouville frequencies. Our approach combines semi-algebraic discrepancy estimates for the Kronecker sequence with quantitative Green's function estimates adapted to the Liouville setting.

Paper Structure

This paper contains 13 sections, 19 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transfer matrix and Lyapunov exponent
  4. Green's Function
  5. Semi-algebraic sets
  6. Discrepancy
  7. Criterion for quantum dynamics
  8. Discrepancy estimates
  9. Large deviation theorem for Liouville frequencies
  10. Proof of main results
  11. Proof of Theorem \ref{['qdDC']}
  12. Proof of Theorem \ref{['qdWDC']}
  13. Proof of Theorem \ref{['qdLiou']}

Key Result

Theorem 1.1

Let $V\in C^{\omega}(\mathbb{T},\mathbb{R})$ be non-constant and assume $L(E)>\lambda>0$ for all $E\in\mathbb{R}$. Let $\eta>0, \gamma\geqslant 1$. Suppose that $\alpha\in \mathbb{R}$ satisfies Then there exists $C_{0}>0$ such that for any $p>0,\varepsilon>0$ and $\phi\in\ell^{2}(\mathbb{Z})$ compactly supported, there exists $T_{1}(\alpha,V,\lambda,\phi,p,\varepsilon)>0$ such that for $T\geqslan

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1: MR2100420
  • Lemma 2.2: MR2100420MR880035MR889980MR3990603
  • Definition 2.3: MR1470456
  • Theorem 2.4: MR1470456
  • Theorem 3.1
  • proof
  • ...and 24 more