Showing papers by "R. A. Serota published in 2012"

Global Level Number Variance in Integrable Systems

Abstract: We study previously un-researched second order statistics - correlation function of spectral staircase and global level number variance - in generic integrable systems with no extra degeneracies. We show that the global level number variance oscillates persistently around the saturation spectral rigidity. Unlike other second order statistics - including correlation function of spectral staircase - which are calculated over energy scales much smaller than the running spectral energy, these oscillations cannot be explained within the diagonal approximation framework of the periodic orbit theory. We give detailed numerical illustration of our results using four integrable systems: rectangular billiard, modified Kepler problem, circular billiard and elliptic billiard.

Quantum mechanical calculation of spectral statistics of a modified Kepler problem.

Abstract: For a modified Kepler problem, we reexamine jumps in the saturation spectral rigidity and large oscillations of the level number variance with near zero minima. Earlier discrepancy between the periodic orbit theory and numerical calculation is cleared by a quantum mechanical calculation. A new class of radial periodic orbits is included establishing a complete correspondence between the periodic orbit theory and the quantum mechanical approach. We show that the diagonal approximation for the level density in the periodic orbit theory already gives a good fit with the numerical calculation. Even greater accuracy is achieved by considering coherent interference between the classical periodic orbits term and the Balian-Bloch term. This procedure produces improved results for the hard-wall rectangular billiards as well.

Level repulsion in integrable systems

Abstract: We provide evidence that level repulsion in semiclassical spectrum is not just a feature of classically chaotic systems, but classically integrable systems as well. While in chaotic systems level repulsion develops on a scale of the mean level spacing, regardless of the location in the spectrum, in integrable systems it develops on a much longer scale — such as geometric mean of the mean level spacing and the running energy in the spectrum for hard wall billiards. We show that at this scale level correlations in integrable systems have a universal dependence on the level separation, as well as discuss their exact form at any scale. These correlations have dramatic consequences, including deviations from the Poissonian statistics in the nearest level spacing distribution and persistent oscillations of the level number variance over an energy interval as a function of the interval width. We illustrate our findings on two specific models — rectangular infinite well and a modified Kepler problem — that serve as generic types of a hard wall billiard and a potential problem without extra symmetries. Our theory and numerical work are based on the concept of parametric averaging that allows sampling of a statistical ensemble of integrable systems at a given spectral location (running energy).