Quantum Bianchi-VII problem, Mathieu functions and arithmetic

TL;DR

This paper investigates the integrability of the geodesic problem on Bianchi-VII$_0$ compact threefolds, providing explicit eigenfunctions via Mathieu functions, and explores spectral statistics in relation to number theory and existing models.

Contribution

It explicitly solves the eigenfunctions of the Laplace-Beltrami operator on Bianchi-VII$_0$ manifolds using Mathieu functions and connects spectral properties with number theory and prior geometric cases.

Findings

The geodesic problem is integrable on Bianchi-VII$_0$ manifolds.

Eigenfunctions are explicitly expressed in terms of Mathieu functions.

Spectral statistics relate to number theory and the Berry-Tabor conjecture.

Abstract

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.