Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension

TL;DR

This paper introduces a method to establish quantum dynamical lower bounds in one-dimensional Schrödinger operators using power-law bounds on transfer matrices, applicable to models like the Fibonacci Hamiltonian.

Contribution

It provides a new approach to quantum dynamical bounds based on transfer matrix estimates for specific energy sets, extending analysis to various models.

Findings

Applicable to Fibonacci Hamiltonian and other models

Establishes quantum dynamical lower bounds from transfer matrix bounds

Simplifies analysis by requiring bounds on a nonempty set of energies

Abstract

We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schr\"odinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.