Quantum Bianchi-VII problem, Mathieu functions and arithmetic

The geodesic problem on the compact threefolds with the Riemannian metric of Bianchi-VII${}_0$ type is studied in both classical and quantum cases. We show that the problem is integrable and describe the eigenfunctions of the corresponding Laplace-Beltrami operators explicitly in terms of the Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. We use the results from number theory to discuss the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation with Bianchi-VI${}_0$ case (Sol-case in Thurston’s classification) and with Bianchi-IX case, corresponding to the classical Euler top.