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

TL;DR

The paper addresses dispersive decay for 1D Dirac equations under time-periodic, nonlocalized forcing, showing that the decay rate can be significantly slower than the autonomous case. By formulating the monodromy map via Fourier integrals and applying stationary-phase analysis, it proves that a mass-switching model generically yields a decay of order n^{-1/3}, with an even slower n^{-1/5} decay at a discrete set of exceptional masses; a rotating-mass variant, however, maps to the autonomous massive Dirac equation and retains the familiar t^{-1/2}-type decay. The methods combine explicit Fourier representations, van der Corput estimates, and Airy-type asymptotics to quantify decay and provide asymptotic expansions tied to initial data. The results illuminate how spatial nonlocality in time-periodic forcing alters energy transport in Floquet-like Dirac systems and have implications for transport in Floquet materials and related wave systems.

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: $i\partial_tα=D(t)α$, where $D(t)=iσ_3\partial_x+ ν(t)$ is time-periodic but not spatially localized. For the special case $ν(t)=mσ_1$, which models a relativistic particle of constant mass $m$, one has a dispersive decay bound: $\|α(t,x)\|_{L^\infty_x}\lesssim t^{-\frac12}$. Previous analyses of Schrödinger Hamiltonians 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 $L^\infty_x$ time-decay rate are considerably slower: $\mathcal{O}(t^{-1/3})$ or even $\mathcal{O}(t^{-1/5})$ as $t\to\infty$.

Figures (3)

• Figure 1: ( A) $\theta ^{(3)}(0)$ as a function of $m>0$, where the dispersion relation $\theta (\xi)$ is given in \ref{['eq:theta']}. ( B) $\theta ^{(3)}(0)$ and $\theta^{(5)}(0)$ overlaid.
• Figure 2: The dispersion relation $\theta (\xi)$, see \ref{['eq:theta']}, for $m=1$. (A)$\theta"(\xi)$ (blue) and $\theta "'(\xi)$ (orange). Each has an increasing sequence of zeroes. (B) Denoting the zeroes of $\theta "$ as $(\xi_l)_{l=1}^{\infty}$, we plot $\theta "'(\xi_l)$ (blue, stars) on a log-log grid, and a polynomial fit (orange, solid) which yields that $|\theta "'(\xi_l)|\lesssim \xi_l ^{-2}$.
• Figure 3: Log-log plot of $\theta^{(5)}(0,m_k)$ which shows that $\theta^{(5)}(0,m_k)\sim m_k^{-3}$.

Theorems & Definitions (14)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Conjecture 2.4
• Theorem 2.5
• Corollary 2.6
• Lemma 5.1
• Lemma 5.2
• proof
• Lemma 5.3