On the spectrum of the Schrödinger operator on 𝕋d: a normal form approach

Abstract In this paper, we study the spectrum of the operator (0.1) on with Γ a maximal dimension lattice in and a pseudodifferential operator of order strictly smaller than M. We prove that most of its eigenvalues admit the asymptotic expansion (0.2) where Z is a function (symbol) and (the dual lattice of Γ).