A note on the RAGE Theorem and phase-averaged dispersion for the Fibonacci Hamiltonian
Gaétan Leclerc
TL;DR
The paper addresses delocalization under weaker spectral hypotheses by introducing eventual absolute continuity, defined through tensor-power convolution of spectral measures, and showing RAGE-type decay for states in the resulting $\mathcal{H}_{eac}$. It then applies this framework to the Fibonacci Hamiltonian, leveraging phase-averaged decay of the density of states to establish that $\mu^{*N}$ becomes absolutely continuous for large $N$, which in turn implies an escape-of-mass phenomenon for averaged evolution. The central technical step is constructing a multi-site Hamiltonian $H_{\vec{\omega}}$ and using Radon–Nikodym arguments to transfer ac spectral regularity to time-averaged matrix elements, enabling decay for $\ell^1$-localized states and arbitrary target vectors. Overall, the work connects spectral measure regularity, tensor-power convolutions, and phase-averaged transport to obtain strong delocalization bounds in a quasi-periodic setting.
Abstract
We find a weaker condition on spectral measures, "eventual absolute continuity", that ensure quantum delocalization as in the RAGE Theorem in the case of purely absolutely continuous spectrum. We then adapt these idea to strongly improve some phase-averaged delocalization bounds for the Fibonacci quasicrystal.