On the Time-decay of solutions arising from periodically forced Dirac Hamiltonians

Abstract

There is increased interest in time-dependent (non-autonomous) Hamiltonians, stemming in part from the active field of Floquet quantum materials. Despite this, dispersive time-decay bounds, which reflect energy transport in such systems, have received little attention. We study the dynamics of non-autonomous, time-periodically forced, Dirac Hamiltonians: $i\partial_t\alpha =D(t)\alpha$, where $D(t)=i\sigma_3\partial_x+ \nu(t)$ is time-periodic but not spatially localized. For the special case $\nu(t)=m\sigma_1$, which models a relativistic particle of constant mass $m$, one has a dispersive decay bound: $\|\alpha(t,x)\|_{L^\infty_x}\lesssim t^{-\frac12}$. Previous analyses of Schr\"odinger Hamiltonians suggest that this decay bound persists for small, spatially-localized and time-periodic $\nu(t)$. However, we show that this is not necessarily the case if $\nu(t)$ is not spatially localized. Specifically, we study two non-autonomous Dirac models whose time-evolution (and monodromy operator) is constructed via Fourier analysis. In a rotating mass model, the dispersive decay bound is of the same type as for the constant mass model. However, in a model with a periodically alternating sign of the mass, the results are quite different. By stationary-phase analysis of the associated Fourier representation, we display initial data for which the $L^\infty_x$ time-decay rate are considerably slower: $\mathcal{O}(t^{-1/3})$ or even $\mathcal{O}(t^{-1/5})$ as $t\to\infty$.