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Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field

Peter D. Hislop, Eric Soccorsi

TL;DR

This work analyzes the large-time edge transport in a time-fractional Schrödinger equation on a half-plane under a constant magnetic field, parameterized by $(\alpha,\beta)$. Using a direct integral fiber decomposition of the magnetic Hamiltonian and Mittag-Leffler evolution, the authors derive explicit expressions for the edge current and establish a transport transition: edge currents grow exponentially when $0<\beta<\alpha$, remain asymptotically constant at $\beta=\alpha$, and decay as $t^{-1-3\alpha}$ when $\alpha<\beta\le1$. The study also computes the mean-square displacement in the $y$-direction, revealing ballistic behavior for the Naber model ($\alpha=\beta$) and decay $t^{-2\alpha}$ in the AYH regime ($\alpha<\beta$). Special cases corresponding to established TFSE models (Naber and AYH) illustrate the framework and confirm the physical relevance of the $\alpha=\beta$ and $\alpha<\beta$ regimes, as discussed by Laskin. Overall, the results quantify how fractional time dynamics alter edge-state transport in quantum Hall-type systems and connect to broader fractional quantum mechanics literature.

Abstract

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.

Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field

TL;DR

This work analyzes the large-time edge transport in a time-fractional Schrödinger equation on a half-plane under a constant magnetic field, parameterized by . Using a direct integral fiber decomposition of the magnetic Hamiltonian and Mittag-Leffler evolution, the authors derive explicit expressions for the edge current and establish a transport transition: edge currents grow exponentially when , remain asymptotically constant at , and decay as when . The study also computes the mean-square displacement in the -direction, revealing ballistic behavior for the Naber model () and decay in the AYH regime (). Special cases corresponding to established TFSE models (Naber and AYH) illustrate the framework and confirm the physical relevance of the and regimes, as discussed by Laskin. Overall, the results quantify how fractional time dynamics alter edge-state transport in quantum Hall-type systems and connect to broader fractional quantum mechanics literature.

Abstract

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 . The TFSE is parameterized by two constants in , where is the fractional order of the time derivative, and is the power of in the Schrodinger equation. We prove that for fixed , there is a transition in the transport properties as varies in : For , the edge current grows exponentially in time, for , the edge current is asymptotically constant, and for , the edge current decays in time. We prove that the mean square displacement in the -direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin \cite{laskin2000_1} that the latter two cases, and , are the physically relevant ones.
Paper Structure (25 sections, 2 theorems, 95 equations)

This paper contains 25 sections, 2 theorems, 95 equations.

Table of Contents

  1. Introduction and statement of the problem
  2. Fractional quantum mechanics
  3. The quantum Hall model
  4. The edge current
  5. Summary of results on the edge current
  6. Outline of the paper
  7. Acknowledgement
  8. The well-posedness of constant-order time-fractional Schrödinger equations
  9. Main results on the TFSE
  10. Proof of Theorem \ref{['thm:existence1']}
  11. Calculation of the edge current
  12. Choice of the initial state $u_0$
  13. Formula for the edge current
  14. Asymptotic behavior of the edge current
  15. Asymptotic expansions of Mittag-Leffler functions
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $(\alpha, \beta) \in (0,1)^2$ be the parameters of the TFSE described in eq1-eq3. We distinguish three cases: In each case, there exists a unique solution $u \in C(\overline{\mathbb{R}^+},L^2(\Omega_0)) \cap C(\mathbb{R}^+,D(H))$ to eq1--eq3 of the form such that The constants $C_j$, $j=1,2,3$ depend on $\alpha, \beta, \Omega$, and $H$.

Theorems & Definitions (5)

  • Theorem 2.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Lemma A.1