{
  "id": "2401.07117",
  "version": "v1",
  "published": "2024-01-13T16:53:09.000Z",
  "updated": "2024-01-13T16:53:09.000Z",
  "title": "Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field",
  "authors": [
    "Peter D. Hislop",
    "Eric Soccorsi"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "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 $(\\alpha, \\beta)$ in $(0,1]$, where $\\alpha$ is the fractional order of the time derivative, and $\\beta$ is the power of $i$ in the Schrodinger equation. We prove that for fixed $\\alpha$, there is a transition in the transport properties as $\\beta$ varies in $(0,1]$: For $0 < \\beta < \\alpha$, the edge current grows exponentially in time, for $\\alpha = \\beta$, the edge current is asymptotically constant, and for $\\beta > \\alpha$, 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, $\\alpha = \\beta$ and $\\alpha < \\beta$, are the physically relevant ones.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-13T16:53:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "35R11",
      "81Q99"
    ],
    "keywords": [
      "constant magnetic field",
      "half-plane",
      "time-fractional schrodinger equations",
      "edge current grows",
      "mean square displacement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}