Time-fractional discrete diffusion equation for Schrödinger 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 Schrödinger operator, ${\mathcal{H}}_{\hslash ,V}:=-{\hslash }^{-2}{\mathcal{L}}_{\hslash }+V$ on the lattice $\hslash {\mathbb{Z}}^n,$ where $V$ is a non-negative multiplication operator and ${\mathcal{L}}_{\hslash }$ 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 $\hslash \to 0$.