Dynamical Upper Bounds for the Fibonacci Hamiltonian

TL;DR

This paper develops a method to bound wavepacket transport in quantum systems and applies it to the Fibonacci Hamiltonian, revealing the exponents are strictly between zero and one at large coupling, confirming theoretical predictions.

Contribution

The paper introduces a general technique for upper bounding transport exponents using transfer matrix norms at complex energies, applied specifically to the Fibonacci Hamiltonian.

Findings

Transport exponents are between 0 and 1 for large coupling.

The large coupling behavior matches predictions by Abe and Hiramoto.

The method provides a unified approach for upper bounds in quantum dynamics.

Abstract

We consider transport exponents associated with the dynamics of a wavepacket in a discrete one-dimensional quantum system and develop a general method for proving upper bounds for these exponents in terms of the norms of transfer matrices at complex energies. Using this method, we prove such upper bounds for the Fibonacci Hamiltonian. Together with the known lower bounds, this shows that these exponents are strictly between zero and one for sufficiently large coupling and the large coupling behavior follows a law predicted by Abe and Hiramoto.