Weighted Weyl estimates near an elliptic trajectory

Abstract

Let ${\psi }_j^{\hslash }$ and $E_j^{\hslash }$ denote the eigenfunctions and eigenvalues of a Schrödinger type operator $H_{\hslash }$ 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 $\hslash ⟶0$ 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 $c$.