An extension of the Floquet-Bloch theory to nilpotent groups and its applications (Recent Developments in Representation Theory and Related Topics)

Abstract

The Floquet-Bloch theory is a popular tool for the investigation of materials with periodic structures. For example, one can show that the spectrum of periodic Schrödinger operators have band structures. In the context of this note, this theory was applied to the following problems in the case of abelian extensions: (1) A geometric analogue of the Chebotarev density theorem for prime closed geodesics in a compact Riemannian manifold with negative curvature (2) A long time asymptotic expansion of the heat kernels of covering manifolds of compact Riemannian manifolds. In this note, we shall develop our version of non-commutative Floquet-Bloch theory and give applications to these problems for nilpotent groups with emphasis on the second topic. Moreover, as a by-product, we give another mathematical explanation of the semi-classical asymptotic expansion formula for the Harper operator due to Wilkinson, which is originally done by Helffer-Sjöstrand.