On the Time-decay of solutions arising from periodically forced Dirac Hamiltonians
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916669235920896 |
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| author | Kraisler, Joseph Sagiv, Amir Weinstein, Michael I. |
| author_facet | Kraisler, Joseph Sagiv, Amir Weinstein, Michael I. |
| contents | 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 $t\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Time-decay of solutions arising from periodically forced Dirac Hamiltonians Kraisler, Joseph Sagiv, Amir Weinstein, Michael I. Analysis of PDEs Mathematical Physics Spectral Theory 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 $t\to\infty$. |
| title | On the Time-decay of solutions arising from periodically forced Dirac Hamiltonians |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2501.07466 |