On correlations of CM-type Maass waveforms under the horocyclic flow

Abstract

Maass waveforms of CM-type are a special kind of eigenfunction of the hyperbolic Laplacian whose 'defining' components (namely eigenvalue and Fourier coefficients) are given by simple formulae involving algebraic integers chosen from a suitable number field K/Q. In this paper, we report on some computer experiments aimed at ascertaining the extent to which the autocorrelation behaviour of CM-forms agrees with that of 'mock' (i.e. random) waveforms in the limit of high energy. Our results suggest that no significant differences are seen.PACS number: 05.45.Mt BackgroundQuantum chaos is a topic of natural interest on negatively curved Riemannian manifolds M. This is a reflection of the well-known fact that the dynamics of a classical point particle traversing M are typically ergodic (cf e.g. [10][11][12]23] and the literature cited there). The simplest class of such manifolds is that associated with the set-up where M is expressed as a quotient space \H , being a discrete subgroup of SL(2, R) and H the Poincaré upper half-plane {Im(z) > 0}. To ensure that \H has good geometry in the hyperbolic metric ds 2 = dx 2 + dy 2 y 2 (the curvature here being −1), one typically assumes that the hyperbolic areaof M is finite. From a dynamical perspective, a good portion of quantum chaos is then concerned with determining the extent to which high-frequency eigenfunctions of the hyperbolic Laplacian on \H simulate random waves as the frequency (or wave number) tends to infinity.

Developments in random matrix theory

“…Wave functions of chaotic quantum systems are discussed in [159]. We also included in this section a paper on the calculation of parametric correlations [160] and a paper on random waves [161].…”