Siegfried Beckus - Quasicrystals and Floquet-Bloch theory, 02nd May 2013

The intention to study aperiodic ordered potentials has risen since Dan Shechtman discovered quasicrystals in the year 1982 (SBGC). In 2011 he has been awarded the Nobel price for his detection. In particular, one is interested to study the long-time behaviour of a corresponding quantum mechanical system. Furthermore, the analysis of the spectrum of such operators is an open question in mathematics. The theory of Schrödinger operators with periodic potentials is well-known as the Floquet-Bloch theory. One uses the so-called Wannier transform to describe such operators by a direct integral. It turns out that the examination of the spectrum on the fibres is a convenient method to obtain results for the spectrum of the Schrödinger operator.
The first part of the talk will focus on the explanation of the definition and properties of the Wannier transform in the aperiodic case. The considerations are mainly based on the preprint "Bloch theory for aperiodic solids" from J. Belissard, G. de Nittis und V. Milani (BNM). The hope is to get some assertions about the spectra of this operators, similar to the periodic case. First the mathematical model of the configuration of such a material will be discussed. Further, we will get to know some essential notions of the Voronoi tiling and the Lagarias group.
Secondly, we will concentrate our considerations on the discrete analogon. For the sake of convenience, we study subshifts in the one-dimensional space where some additional issues arise. In order to handle this, we will discuss the notion of associated graph sequence corresponding to a subshift and the Benjamin-Schramm convergence.


 BNM         
J. Bellissard, G. de Nittis,V. Milani, Bloch theory for aperiodic solids, In preperation.
 SBGC D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Metallic Phase with Long-Range Orientational, The American Physical Society, 1984.