Weyl asymptotics for pseudodifferential operators in a discrete setting

We prove a sharp Weyl estimate for the number of eigenvalues belonging to a fixed interval of energy of a self-adjoint difference operator acting on ${\mathrm{\ell }}^2(\epsilon {\mathbb{Z}}^d)$ if the associated symplectic volume of phase space in ${\mathbb{R}}^d\times {\mathbb{T}}^d$ accessible for the Hamiltonian flow of the principal symbol is finite. Here $\epsilon$ is a semiclassical parameter. Our proof depends crucially on the construction of a good semiclassical approximation for the time evolution induced by the self-adjoint operator on ${\mathrm{\ell }}^2(\epsilon {\mathbb{Z}}^d)$. This extends previous semiclassical results to a broad class of difference operators on a scaled lattice.