Spectral theta series of operators with periodic bicharacteristic flow

[Série thêta spectrale d’opérateurs dont le flot bicaractéristique est périodique]

Marklof, Jens

Annales de l'Institut Fourier, Tome 57 (2007), p. 2401-2427 / Harvested from Numdam

Résumé

La série thêta $\vartheta (z)=\sum \exp (2\pi \mathrm{i} n^2 z)$ est un exemple classique de forme modulaire. Dans cet article, nous montrons que la trace $\vartheta _P(z)= \operatorname{Tr}\,\exp (2\pi \mathrm{i} P^2 z)$, où $P$ est un opérateur pseudo-différentiel elliptique auto-adjoint d’ordre 1 à flot bicaractéristique périodique, en est une généralisation naturelle. En particulier, nous établissons des égalités fonctionnelles approchées sous l’action du groupe modulaire. Ceci permet une analyse détaillée de l’asymptotique de $\vartheta _P(z)$ au voisinage de l’axe réel, et prouve des lois du logarithme et des théorèmes limites pour la distribution de ses valeurs. Ces asymptotiques diffèrent de celles relatives à la trace de l’opérateur des ondes $\operatorname{Tr}\, \exp (-\mathrm{i} P t)$, dont les singularités sont portées par les longueurs des bicaractéristiques périodiques.

Abstract

The theta series $\vartheta (z)=\sum \exp (2\pi \mathrm{i} n^2 z)$ is a classical example of a modular form. In this article we argue that the trace $\vartheta _P(z)=\operatorname{Tr}\,\exp (2\pi \mathrm{i} P^2 z)$, where $P$ is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of $\vartheta _P(z)$ near the real axis, and the proof of logarithm laws and limit theorems for its value distribution. These asymptotics are in fact distinctly different from those for the ‘wave trace’ $\operatorname{Tr}\, \exp (-\mathrm{i} P t)$ whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.

Détail

Publié le : 2007-01-01

MR 2394547 | Zbl 1133.35075

DOI : https://doi.org/10.5802/aif.2338
Classification:  35P20,  11F72,  37D40
Mots clés: série théta spectrale, variété de Zoll, flot géodésique périodique, représentation de Shale-Weil, flot horocyclique, lois du logarithme

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