# On the spectrum of the Schr\"odinger operator on $\mathbb{T}^d$: a normal form approach

## TL;DR

This paper analyzes the spectrum of a fractional Schrödinger operator on a torus, showing that most eigenvalues have a precise asymptotic expansion involving a smooth symbol, using a normal form approach.

## Contribution

It introduces a normal form method to derive asymptotic eigenvalue expansions for fractional Schrödinger operators on tori, extending spectral analysis techniques.

## Key findings

- Most eigenvalues admit a detailed asymptotic expansion
- The eigenvalue expansion includes a smooth symbol function
- The approach applies to operators with pseudodifferential perturbations

## Abstract

In this paper we study the spectrum of the operator \begin{equation} \label{ope} H:=(-\Delta)^{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} \lambda_\xi=|\xi|^M+Z(\xi)+O(\left|\xi\right|^{-\infty})\ , \end{equation} where $Z$ is a $C^\infty(\mathbb{R}^d)$ function (symbol) and $\xi\in\Gamma^*$ (the dual lattice of $\Gamma$).