David Aristoff's homepage Title: Power Spectrum Analysis in Quantum Chaology

Abstract: By the Bohigas-Giannoni-Schmit conjecture (1984), the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behavior are described by random matrix theory. An alternative characterization of eigenvalue fluctuations was suggested where a long sequence of eigenlevels has been interpreted as a discrete-time random process. It has been conjectured that the power spectrum of energy level fluctuations shows 1/ω noise in the chaotic case, whereas, when the classical analog is fully integrable, it shows 1/ω2 behavior. This is expected for frequencies 1<<ω << n where n is the number of eigenlevels. We consider the power spectrum of the circular unitary ensemble with an additional fixed charge at 1. We will show that when the frequency gets of order n there is a correction to the 1/ω law which is described by a Painlev ́e V equation. Further we show a relation to Toeplitz determinants.