Time-fractional discrete diffusion equation for Schrödinger operator

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 Schr & ouml;dinger operator, H & hstrok;,V:=-& hstrok;-2L & hstrok;+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_{\hbar ,V}:=-\hbar <^>{-2}\mathcal {L}_{\hbar }+V$$\end{document} on the lattice & hstrok;Zn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \mathbb {Z}<^>{n},$$\end{document} where V is a positive multiplication operator and L & hstrok;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\hbar }$$\end{document} 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 & hstrok;-> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \rightarrow 0$$\end{document}.