Level Spacings for Integrable Quantum Maps in Genus Zero

TL;DR

This paper investigates the eigenvalue spacings of certain integrable quantum maps on the 2-sphere, demonstrating that for most cases, the spacings tend to follow a Poisson distribution as the system size grows.

Contribution

It proves a weak form of the Berry-Tabor conjecture for a broad class of integrable quantum maps, showing Poissonian eigenvalue spacing distribution for almost all parameters.

Findings

Eigenvalue spacings tend to Poisson distribution for almost all parameters.

The result holds along a sparse subsequence of system sizes.

The proof supports the Berry-Tabor conjecture in a specific quantum map setting.

Abstract

We consider the eigenvalue pair correlation problem for certain integrable quantum maps on the 2-sphere. The classical maps are time one maps of Hamiltonian flows of perfect Morse functions. The quantizations are unitary operators on spaces of homogeneous holomorphic polynomials of degree N in two complex variables. There are N eigenphases on the unit circle and the pair correlation problem is to determine the distribution of spacings between the eigenphases on the length scale of the mean level spacing. The Berry-Tabor conjecture says that for generic integrable systems, the spacings distribution should tend to the Poisson (uniform) limit as N tends to infinity. In this paper and in its addendum we prove a somewhat weak form of this conjecture: We define a large class of two parameter families of integrable Hamiltonians, and prove that for almost all elements of each family the limit pair correlation function is Poisson. In this paper, we take the limit along a slightly sparse subsequence of N's; in the addendum we take Cesaro means along the full seuqence.