Operators on Abelian Coverings of Compact Manifolds

This truly short chapter indicates the possibility of translating a large number of techniques and some results to the case of periodic operators on abelian (i.e., commutative) coverings of compact manifolds. The “usual” periodic case corresponds to the natural covering of the n-dimensional torus by the n-dimensional Euclidean space. While most of the Floquet theory technique transfers easily to this more general case, some unresolved questions and conjectures still remain. An example is the Sunada’s conjecture of absence of spectral gaps in the spectrum of the Laplace–Beltrami operator on the maximal abelian covering if the base is of constant negative curvature.