# Spectral theta series of operators with periodic bicharacteristic flow

Jens Marklof^{[1]}

- [1] University of Bristol School of Mathematics Bristol BS8 1TW (United Kingdom)

Annales de l’institut Fourier (2007)

- Volume: 57, Issue: 7, page 2401-2427
- ISSN: 0373-0956

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topMarklof, Jens. "Spectral theta series of operators with periodic bicharacteristic flow." Annales de l’institut Fourier 57.7 (2007): 2401-2427. <http://eudml.org/doc/10302>.

@article{Marklof2007,

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.},

affiliation = {University of Bristol School of Mathematics Bristol BS8 1TW (United Kingdom)},

author = {Marklof, Jens},

journal = {Annales de l’institut Fourier},

keywords = {Spectral theta series; Zoll manifolds; periodic geodesic flow; Shale-Weil representation; horocycle flow; logarithm laws; spectral theta series},

language = {eng},

number = {7},

pages = {2401-2427},

publisher = {Association des Annales de l’institut Fourier},

title = {Spectral theta series of operators with periodic bicharacteristic flow},

url = {http://eudml.org/doc/10302},

volume = {57},

year = {2007},

}

TY - JOUR

AU - Marklof, Jens

TI - Spectral theta series of operators with periodic bicharacteristic flow

JO - Annales de l’institut Fourier

PY - 2007

PB - Association des Annales de l’institut Fourier

VL - 57

IS - 7

SP - 2401

EP - 2427

AB - 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.

LA - eng

KW - Spectral theta series; Zoll manifolds; periodic geodesic flow; Shale-Weil representation; horocycle flow; logarithm laws; spectral theta series

UR - http://eudml.org/doc/10302

ER -

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