An extension of the Bloch-Floquet theory to the Heisenberg group and its applications to asymptotic problems for heat kernels and prime closed geodesics (Mathematical aspects of quantum fields and related topics)

Abstract

The Bloch-Floquet theory are popular tools for the investigation of materials with periodic structures. For example, we can show that the spectrum of periodic Schrodinger operators have band structures. Here we shall extend the Bloch-Floquet theory, which is appicable to abelian groups, to the Heisenberg group. Our method is based on a combination of the representations of the discrete Heisenberg groups and of the Heisenberg Lie group. We apply this method to asymptotic problems for heat kernels and for counting prime closed geodesics. In this application, we need additional ingradients, the semi-classical analysis and the Chen's iterated integrals. As a by-product, we give another mathematical explanation of the semi-classical asymptotic expansion formula for the Hofstadter butterfly of Wilkinson, which is originally due to Helffer-Sjostrand.