Time-fractional discrete diffusion equation for Schrödinger operator

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Aparajita Dasgupta, Shyam Swarup Mondal, Michael Ruzhansky, Abhilash Tushir
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引用次数: 0

Abstract

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, \(\mathcal {H}_{\hbar ,V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) on the lattice \(\hbar \mathbb {Z}^{n},\) where V is a positive multiplication operator and \(\mathcal {L}_{\hbar }\) is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit \(\hbar \rightarrow 0\).

薛定谔算子的时间分数离散扩散方程
本文旨在研究与离散薛定谔算子(\(\mathcal {H}_{\hbar ,V}.=-\hbar^{-2}\mathcal {L}_{\hbar }+V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) on the lattice \(\hbar \mathbb {Z}^{n},\) where V is a positive multiplication operator and \(\mathcal {L}_{\hbar }\) is the discrete Laplacian.我们在相关的 Sobolev 型空间中建立了具有规则系数的一般 Caputo 型扩散方程的 Cauchy 问题的良好求解性。然而,当扩散系数具有分布奇异性时,该问题的良好求解程度很弱。最后,我们在半经典极限 \(\hbar\rightarrow 0\) 中重获了一般卡普托型扩散方程的经典解(即非常弱的解)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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