Spectral theta series of operators with periodic bicharacteristic flow

Marklof, Jens

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2401-2427.

La série thêta $\vartheta \left(z\right)=\sum exp\left(2\pi \mathrm{i}{n}^{2}z\right)$ est un exemple classique de forme modulaire. Dans cet article, nous montrons que la trace ${\vartheta }_P\left(z\right)=Trexp\left(2\pi \mathrm{i}{P}^{2}z\right)$, 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 ${\vartheta }_P\left(z\right)$ 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 $Trexp(-\mathrm{i}Pt)$, dont les singularités sont portées par les longueurs des bicaractéristiques périodiques.

The theta series $\vartheta \left(z\right)=\sum exp\left(2\pi \mathrm{i}{n}^{2}z\right)$ is a classical example of a modular form. In this article we argue that the trace ${\vartheta }_P\left(z\right)=Trexp\left(2\pi \mathrm{i}{P}^{2}z\right)$, 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 ${\vartheta }_P\left(z\right)$ 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’ $Trexp(-\mathrm{i}Pt)$ whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.

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