# Exact dynamical decay rate for the almost Mathieu operator

## TL;DR

This paper proves that for supercritical almost Mathieu operators with Diophantine frequencies, the exponential decay rate in expectation equals the Lyapunov exponent, establishing a precise relationship in their spectral properties.

## Contribution

It establishes that the exponential decay rate in expectation is well defined and equals the Lyapunov exponent for a class of almost Mathieu operators, clarifying their spectral decay characteristics.

## Key findings

- Exponential decay rate in expectation equals Lyapunov exponent.
- Decay rate is well defined for supercritical almost Mathieu operators.
- Results apply to operators with Diophantine frequencies.

## Abstract

We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies.