Time-fractional discrete diffusion equation for Schr\"{o}dinger operator

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 Schr\"{o}dinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $V$ is a non-negative 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 $\hbar\to 0$.