Spectra of Sol-Manifolds: Arithmetic and Quantum Monodromy

The spectral problem of three-dimensional manifolds M³_A admitting Sol-geometry in Thurston's sense is investigated. Topologically M³_A are torus bundles over a circle with a unimodular hyperbolic gluing map A. The eigenfunctions of the corresponding Laplace-Beltrami operators are described in terms of modified Mathieu functions. It is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form. As a result the spectral statistics is shown to disagree with the Berry-Tabor conjecture. The topological nature of the monodromy for both classical and quantum systems on Sol-manifolds is demonstrated.