# Weighted Weyl estimates near an elliptic trajectory

### Thierry Paul

Ecole Polytechnique, Palaiseau, France### Alejandro Uribe

University of Michigan, Ann Arbor, USA

## Abstract

Let $ψ_{j}$ and $E_{j}$ denote the eigenfunctions and eigenvalues of a Schrödinger type operator $H_{ℏ}$ with discrete spectrum. Let $ψ_{(x,ξ)}$ be a coherent state centered at a point $(x,ξ)$ belonging to an elliptic periodic orbit, $γ$ of action $Sγ$ and Maslov index $σ_{γ}$. We consider “weighted Weyl estimates” of the following form: we study the asymptotics, as $ℏ⟶0$ along any sequence

$l∈N,α∈R$ fixed, of

We prove that the asymptotics depend strongly on $α$-dependent arithmetical properties of $c$ and on the angles $θ$ of the Poincaré mapping of $γ$. In particular, under irrationality assumptions on the angles, the limit exists for a non-open set of full measure of $c$'s. We also study the regularity of the limit as a function of $c$.

## Cite this article

Thierry Paul, Alejandro Uribe, Weighted Weyl estimates near an elliptic trajectory. Rev. Mat. Iberoam. 14 (1998), no. 1, pp. 145–165

DOI 10.4171/RMI/238