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

In this paper, we study the spectrum of the operator (Formula presented.) on L2(Rd/Г) with Γ a maximal dimension lattice in Rd and V a pseudodifferential operator of order strictly smaller than M. We prove that most of its eigenvalues admit the asymptotic expansion (Formula presented.) where Z is a C∞(Rd) function (symbol) and ξϵГ* (the dual lattice of Γ).