Asymptotic analysis of time-fractional quantum diffusion

Peter D. Hislop; Éric Soccorsi

doi:10.1016/j.aml.2024.109033

Asymptotic analysis of time-fractional quantum diffusion

Peter D. Hislop
, Éric Soccorsi

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrödinger equation in R^d. We define the time-fractional derivative by the Caputo derivative. We consider the initial-value problem for the free evolution of wave packets in R^d governed by the time-fractional Schrödinger equation i^β∂_t^αu=−Δu, with initial condition u(t=0)=u₀, parameterized by two indices α,β∈(0,1]. We show distinctly different long-time evolution of the mean square displacement according to the relation between α and β. In particular, asymptotically ballistic motion occurs only for α=β.

Original language	English
Article number	109033
Journal	Applied Mathematics Letters
Volume	152
DOIs	https://doi.org/10.1016/j.aml.2024.109033
State	Published - Jun 2024

Bibliographical note

Publisher Copyright:
© 2024 Elsevier Ltd

Funding

The authors thank Yavar Kian for discussions on the topic of this paper. PDH thanks Aix Marseille Université for some financial support and hospitality during the time parts of this paper were written. PDH is partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 843327 . ÉS is partially supported by the Agence Nationale de la Recherche (ANR) under grant ANR-17-CE40-0029 . The authors thank Yavar Kian for discussions on the topic of this paper. PDH thanks Aix Marseille Université for some financial support and hospitality during the time parts of this paper were written. PDH is partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 843327. ÉS is partially supported by the Agence Nationale de la Recherche (ANR) under grant ANR-17-CE40-0029.

Funders	Funder number
Simons Foundation Collaboration Grant for Mathematicians	843327
Aix-Marseille Université
French Agence Nationale de la Recherche	ANR-17-CE40-0029

Keywords

Quantum diffusion
Schrödinger equation
Time-fractional derivative

ASJC Scopus subject areas

Applied Mathematics

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10.1016/j.aml.2024.109033

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title = "Asymptotic analysis of time-fractional quantum diffusion",

abstract = "We study the large-time asymptotics of the mean-square displacement for the time-fractional Schr{\"o}dinger equation in Rd. We define the time-fractional derivative by the Caputo derivative. We consider the initial-value problem for the free evolution of wave packets in Rd governed by the time-fractional Schr{\"o}dinger equation iβ∂tαu=−Δu, with initial condition u(t=0)=u0, parameterized by two indices α,β∈(0,1]. We show distinctly different long-time evolution of the mean square displacement according to the relation between α and β. In particular, asymptotically ballistic motion occurs only for α=β.",

keywords = "Quantum diffusion, Schr{\"o}dinger equation, Time-fractional derivative",

author = "Hislop, \{Peter D.\} and {\'E}ric Soccorsi",

note = "Publisher Copyright: {\textcopyright} 2024 Elsevier Ltd",

year = "2024",

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doi = "10.1016/j.aml.2024.109033",

language = "English",

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AU - Soccorsi, Éric

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N2 - We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrödinger equation in Rd. We define the time-fractional derivative by the Caputo derivative. We consider the initial-value problem for the free evolution of wave packets in Rd governed by the time-fractional Schrödinger equation iβ∂tαu=−Δu, with initial condition u(t=0)=u0, parameterized by two indices α,β∈(0,1]. We show distinctly different long-time evolution of the mean square displacement according to the relation between α and β. In particular, asymptotically ballistic motion occurs only for α=β.

AB - We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrödinger equation in Rd. We define the time-fractional derivative by the Caputo derivative. We consider the initial-value problem for the free evolution of wave packets in Rd governed by the time-fractional Schrödinger equation iβ∂tαu=−Δu, with initial condition u(t=0)=u0, parameterized by two indices α,β∈(0,1]. We show distinctly different long-time evolution of the mean square displacement according to the relation between α and β. In particular, asymptotically ballistic motion occurs only for α=β.

KW - Quantum diffusion

KW - Schrödinger equation

KW - Time-fractional derivative

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UR - https://www.scopus.com/pages/publications/85186602233#tab=citedBy

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SN - 0893-9659

VL - 152

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

M1 - 109033

ER -

Asymptotic analysis of time-fractional quantum diffusion

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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