The central object of study are Schrodinger operators with potentials displaying aperiodic order. In one dimension there have been recent advances in the understanding of their spectral and quantum dynamical properties, particularly in the case of the Fibonacci potential and related models, so-called Sturmian potentials, which are the standard models of one-dimensional quasicrystals. It is the goal of the proposed research to extend the theory to larger classes of potentials in one dimension and to tackle the higher dimensional case. A crucial tool in one dimension is the trace map, an energy-indexed dynamical system which can be used to characterize and study the spectrum of the operators. Along with combinatorial partition results and Gordon-type criteria one can obtain good bounds on generalized eigenfunctions from which one can deduce spectral and quantum dynamical consequences. It appears feasible that this approach is applicable to potentials beyond the class of Sturmian potentials -- sufficiently low complexity should suffice to induce partitions and trace maps. In higher dimensions the main goal is to find an analog of Gordon's criterion which can serve as a link between combinatorics and spectral theory.
The mathematics of aperiodic order is a young emerging field that has sparked a lot of research activity since the mid-nineties. Researchers from disciplines as diverse as spectral theory, group theory, dynamical systems, combinatorics, and algebraic topology have found a common ground that was motivated by the discovery of quasicrystals in 1984 and the subsequent reconsideration of the nature of order and ordered structures. By now, quite a number of structural models for quasicrystals have been proposed. Joint efforts are being undertaken to investigate their properties and shed light on why quasicrystals exist, how they form, why they are stable. Regarding their electronic transport properties, it is expected that quasicrystals may exhibit anomalous behavior. It is therefore planned to study transport properties of Sturmian and related models from this perspective.
