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, \(\mathcal {H}_{\hbar ,V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) 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 \(\hbar \rightarrow 0\).