On the spectrum of the Schrödinger operator on ${\mathbb{T}}^d$: a normal form approach

Abstract

In this paper we study the spectrum of the operator \begin{equation} \label{ope} H:=(-Δ)^{M/2}+\mathcal{V}\ , \quad M>0\ , \end{equation} on $L^2({\mathbb{R}}^d/\Gamma )$, with $\Gamma$ a maximal dimension lattice in ${\mathbb{R}}^d$ and $\mathcal{V}$ a pseudodifferential operator of order strictly smaller than $M$. We prove that most of its eigenvalues admit the asymptotic expansion \begin{equation} \label{sim} λ_ξ=|ξ|^M+Z(ξ)+O(\left|ξ\right|^{-\infty})\ , \end{equation} where $Z$ is a $C^{\infty }({\mathbb{R}}^d)$ function (symbol) and $\xi \in {\Gamma }^{\ast }$ (the dual lattice of $\Gamma$).