Quantum Dynamical Bounds for Quasi-Periodic Operators with Liouville Frequencies

TL;DR

This paper derives quantum dynamical upper bounds for quasi-periodic Schrödinger operators with Liouville frequencies, using advanced discrepancy and Green's function estimates to handle the complex frequency properties.

Contribution

It introduces a novel method combining discrepancy estimates and Green's function analysis to address Liouville frequency challenges in quantum dynamics.

Findings

Established quantum dynamical upper bounds for Liouville frequency operators

Developed semi-algebraic discrepancy estimates for Kronecker sequences

Adapted Green's function estimates to the Liouville setting

Abstract

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