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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2401.07117 |
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| _version_ | 1866917567248990208 |
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| author | Hislop, Peter D. Soccorsi, Eric |
| author_facet | Hislop, Peter D. Soccorsi, Eric |
| contents | We study the large-time asymptotics of the edge current for a family of time-fractional Schrodinger equations with a constant, transverse magnetic field on a half-plane $(x,y) \in \mathbb{R}_x^+ \times \mathbb{R}_y$. The TFSE is parameterized by two constants $(α, β)$ in $(0,1]$, where $α$ is the fractional order of the time derivative, and $β$ is the power of $i$ in the Schrodinger equation. We prove that for fixed $α$, there is a transition in the transport properties as $β$ varies in $(0,1]$: For $0 < β< α$, the edge current grows exponentially in time, for $α= β$, the edge current is asymptotically constant, and for $β> α$, the edge current decays in time. We prove that the mean square displacement in the $y\in \mathbb{R}$-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin \cite{laskin2000_1} that the latter two cases, $α= β$ and $α< β$, are the physically relevant ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field Hislop, Peter D. Soccorsi, Eric Mathematical Physics 35Q40 35R11 81Q99 We study the large-time asymptotics of the edge current for a family of time-fractional Schrodinger equations with a constant, transverse magnetic field on a half-plane $(x,y) \in \mathbb{R}_x^+ \times \mathbb{R}_y$. The TFSE is parameterized by two constants $(α, β)$ in $(0,1]$, where $α$ is the fractional order of the time derivative, and $β$ is the power of $i$ in the Schrodinger equation. We prove that for fixed $α$, there is a transition in the transport properties as $β$ varies in $(0,1]$: For $0 < β< α$, the edge current grows exponentially in time, for $α= β$, the edge current is asymptotically constant, and for $β> α$, the edge current decays in time. We prove that the mean square displacement in the $y\in \mathbb{R}$-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin \cite{laskin2000_1} that the latter two cases, $α= β$ and $α< β$, are the physically relevant ones. |
| title | Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field |
| topic | Mathematical Physics 35Q40 35R11 81Q99 |
| url | https://arxiv.org/abs/2401.07117 |