Quantum Bianchi-VII problem, Mathieu functions and arithmetic (Q6156144)

This paper is devoted to the geodesic problem on compact threefolds with Riemannian metric of Bianchi-type. This problem is studied in both classical and quantum cases. The authors show that the problem is integrable; they describe explicitly the eigenfunctions of the corresponding Laplace-Beltrami operators in terms of Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. They employ a number-theoretic approach in order to study the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation when some Bianchi-type metrics are taken into account.