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

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: [Figure presented], where [Figure presented] is time-periodic but not spatially localized. For the special case ν(t)=mσ₁, which models a relativistic particle of constant mass m, one has a dispersive decay bound: ‖α(t,x)‖_Lx^∞≲t^{−[Formula presented]}. Previous analyses of Schrödinger Hamiltonians (e.g. [4,29,30,45]) suggest that this decay bound persists for small, spatially-localized and time-periodic ν(t). However, we show that this is not necessarily the case if ν(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_x^∞ time-decay rate are considerably slower: O(t^−1/3) or even O(t^−1/5) as t→∞.