Spectra of SolManifolds: Arithmetic and Quantum Monodromy
Abstract
The spectral problem of threedimensional manifolds M^{3}_{A} admitting Solgeometry in Thurston's sense is investigated. Topologically M^{3}_{A} are torus bundles over a circle with a unimodular hyperbolic gluing map A. The eigenfunctions of the corresponding LaplaceBeltrami 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 BerryTabor conjecture. The topological nature of the monodromy for both classical and quantum systems on Solmanifolds is demonstrated.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 2006
 DOI:
 10.1007/s0022000615436
 arXiv:
 arXiv:mathph/0503046
 Bibcode:
 2006CMaPh.264..583B
 Keywords:

 Neural Network;
 Manifold;
 Statistical Physic;
 Complex System;
 Nonlinear Dynamics;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Mathematics  Number Theory;
 Mathematics  Spectral Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 58J50;
 35P20
 EPrint:
 28 pages, 8 figures