Level spacings for integrable quantum maps in genus zero (Q1265728)

The pair correlation functions (PCFs) of completely integrable quantum maps over \(\mathbb{C} P^1\) are considered. For any 2-parameter family of Hamiltonians of the form \(H_{\alpha,\beta}= \alpha \Phi(\widehat{I})+ \beta \widehat{I}\) the author proves that this conjecture is correct for almost all \((\alpha, \beta)\) along the subsequence of Planck constants \(N_m= [m[\log m)^5]\). In this paper the author compares the results with those of Rudnick-Sarnak on the PCF of fractional parts of polynomials. For the addendum [ibid. 196, 319-329 (1998) see Zbl 0949.37041].