Spectra of Sol-manifolds: arithmetic and quantum monodromy

TL;DR

This paper analyzes the spectral properties of Sol-manifolds, revealing their eigenvalue multiplicities relate to quadratic forms and showing their spectral statistics differ from classical conjectures, with implications for quantum monodromy.

Contribution

It provides a detailed spectral analysis of Sol-manifolds, linking eigenvalue multiplicities to quadratic forms and exploring the topological quantum monodromy.

Findings

Eigenvalue multiplicities relate to representations of integers by quadratic forms.

Spectral statistics do not follow the Berry-Tabor conjecture.

Topological quantum monodromy is demonstrated for Sol-manifolds.

Abstract

The spectral problem of three-dimensional manifolds M_A admitting Sol-geometry in Thurston's sense is investigated. Topologically M_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 the 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.