Quantum Dynamics via Complex Analysis Methods: General Upper Bounds Without Time-Averaging and Tight Lower Bounds for the Strongly Coupled Fibonacci Hamiltonian

TL;DR

This paper advances complex analysis techniques to establish precise upper and lower bounds on quantum dynamics, notably for the Fibonacci Hamiltonian, revealing that time-averaged spreading rates can surpass spectral dimension limits.

Contribution

We extend complex analysis methods to derive bounds on quantum transport, including the first demonstration of exceeding spectral dimension in time-averaged spreading rates for the Fibonacci Hamiltonian.

Findings

Derived upper bounds for non-time averaged probabilities and moments.

Established improved lower bounds for time-averaged transport exponents.

Showed that spreading rates can exceed the spectrum's box-counting dimension.

Abstract

We develop further the approach to upper and lower bounds in quantum dynamics via complex analysis methods which was introduced by us in a sequence of earlier papers. Here we derive upper bounds for non-time averaged outside probabilities and moments of the position operator from lower bounds for transfer matrices at complex energies. Moreover, for the time-averaged transport exponents, we present improved lower bounds in the special case of the Fibonacci Hamiltonian. These bounds lead to an optimal description of the time-averaged spreading rate of the fast part of the wavepacket in the large coupling limit. This provides the first example which demonstrates that the time-averaged spreading rates may exceed the upper box-counting dimension of the spectrum.