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Asymptotic analysis of time-fractional quantum diffusion

Peter D. Hislop, Eric Soccorsi

Abstract

We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrodinger equation in $\mathbb{R}^d$. We define the time-fractional derivative by the Caputo derivative and we consider the initial-value problem for the free evolution of wave packets in $\mathbb{R}^d$ governed by the time-fractional Schrodinger equation $ i^β\partial_t^αu = - Δu, ~~~~u(t=0) = u_0$, parameterized by two indices $α, β\in (0,1]$. 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 $α=β$.

Asymptotic analysis of time-fractional quantum diffusion

Abstract

We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrodinger equation in . We define the time-fractional derivative by the Caputo derivative and we consider the initial-value problem for the free evolution of wave packets in governed by the time-fractional Schrodinger equation , parameterized by two indices . We show distinctly different long-time evolution of the mean square displacement according to the relation between and . In particular, asymptotically ballistic motion occurs only for .
Paper Structure (10 sections, 3 theorems, 59 equations)

This paper contains 10 sections, 3 theorems, 59 equations.

Table of Contents

  1. Introduction and statement of the problem
  2. Acknowledgement
  3. Well-posedness of the TFSE
  4. Existence and uniqueness result
  5. Proof of Proposition \ref{['pr1']}
  6. Diffusion properties of the TFSE
  7. Mean square displacement
  8. Time-fractional quantum diffusion
  9. Settings
  10. Asymptotics

Key Result

Proposition 2.1

Pick $u_0 \in {\mathcal{U}}_{\alpha,\beta}$. Then, for all $T \in {\mathbb R}_+$, the system eq1-eq2 admits a unique solution $u \in {\mathcal{C}}([0,T],{\rm D}(H_0)) \cap W_{\rm loc}^{1,1}(0,T;L^2({\mathbb R}^d)),$ which is expressed by and there exists a unique positive constant $C$, depending only on $\alpha$, $\beta$ and $d$, such that we have for $\beta \ge \alpha$, whereas for $\beta < \a

Theorems & Definitions (6)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof