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Revista Matemática Iberoamericana

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Volume 14, Issue 1, 1998, pp. 145–165
DOI: 10.4171/RMI/238

Published online: 1998-04-30

Weighted Weyl estimates near an elliptic trajectory

Thierry Paul[1] and Alejandro Uribe[2]

(1) Ecole Polytechnique, Palaiseau, France
(2) University of Michigan, Ann Arbor, USA

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 $$\hbar = \frac {S_\gamma} {2\pi l – \alpha + \sigma_\gamma},$$ $l \in \mathbb N, \alpha \in \mathbb R$ fixed, of $$\sum_{|Ej–E|≤c\hbar} | (\psi_{(x, \xi)}, \psi^h_j) |^2.$$ 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 $c$.

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Paul Thierry, Uribe Alejandro: Weighted Weyl estimates near an elliptic trajectory. Rev. Mat. Iberoam. 14 (1998), 145-165. doi: 10.4171/RMI/238