Addendum: Level Spacings for Integrable Quantum Maps in Genus Zero

In this addendum we strengthen the results of math-ph/0002010 in the case of polynomial phases. We prove that Cesaro means of the pair correlation functions of certain integrable quantum maps on the 2-sphere at level N tend almost always to the Poisson (uniform limit). The quantum maps are exponentials of Hamiltonians which have the form a p(I) + b I, where I is the action, where p is a polynomial and where a,b are two real numbers. We prove that for any such family and for almost all a,b, the pair correlation tends to Poisson on average in N. The results involve Weyl estimates on exponential sums and new metric results on continued fractions. They were motivated by a comparison of the results of math-ph/0002010 with some independent results on pair correlation of fractional parts of polynomials by Rudnick-Sarnak.Comment: Addendum to math-ph/000201

Level Spacings for Integrable Quantum Maps in Genus Zero

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