# Weighted Weyl estimates near an elliptic trajectory

### Thierry Paul

Ecole Polytechnique, Palaiseau, France### Alejandro Uribe

University of Michigan, Ann Arbor, USA

## Abstract

Let $\psi^\hbar_j$ and $E^\hbar_j$ denote the eigenfunctions and eigenvalues of a Schrödinger type operator $H_\hbar$ with discrete spectrum. Let $\psi_{(x, \xi)}$ be a coherent state centered at a point $(x, \xi)$ belonging to an elliptic periodic orbit, $\gamma$ of action $S\gamma$ and Maslov index $\sigma_\gamma$. We consider "weighted Weyl estimates" of the following form: we study the asymptotics, as $\hbar \longrightarrow 0$ along any sequence

$l \in \mathbb N, \alpha \in \mathbb R$ fixed, of

We prove that the asymptotics depend strongly on $\alpha$-dependent arithmetical properties of $c$ and on the angles $\theta$ of the Poincaré mapping of $\gamma$. 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