E(k,L) level statistics of classically integrable quantum systems based on the Berry-Robnik approach

TL;DR

This paper extends the Berry-Robnik approach to derive the E(K,L) level statistics function for classically integrable quantum systems, revealing non-Poissonian behaviors in energy level distributions.

Contribution

It introduces a novel derivation of the E(K,L) function for integrable systems, expanding the statistical tools beyond traditional LSD and LNV measures.

Findings

Derived the limiting E(K,L) function for infinitely many components.

Showed deviations from Poisson statistics in energy level distributions.

Provided a fundamental measure for quantum level statistics.

Abstract

Theory of the quantal level statistics of classically integrable system, developed by Makino et al. in order to investigate the non-Poissonian behaviors of level-spacing distribution (LSD) and level-number variance (LNV)\cite{MT03,MMT09}, is successfully extended to the study of $E(K,L)$ function which constitutes a fundamental measure to determine most statistical observables of quantal levels in addition to LSD and LNV. In the theory of Makino et al., the eigenenergy level is regarded as a superposition of infinitely many components whose formation is supported by the Berry-Robnik approach in the far semiclassical limit\cite{Robn1998}. We derive the limiting $E(K,L)$ function in the limit of infinitely many components and elucidates its properties when energy levels show deviations from the Poisson statistics.