Spectral theta series of operators with periodic bicharacteristic flow

Abstract

The theta series script V sign(z) = σexp(2πin2z) is a classical example of a modular form. In this article we argue that the trace script V signp(z) = Tr exp(27πiP2z), 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 script V sign 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' Tr exp(-iPt) whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.