Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals
For mathematicians and physicists studying quasicrystals, this is an incremental survey of known results without novel contributions.
This survey reviews spectral properties of Schrödinger operators in quasicrystal models, focusing on the Fibonacci Hamiltonian and Sturmian potentials. It presents rigorous results and numerical calculations, but does not introduce new findings or quantitative improvements.
We survey results that have been obtained for self-adjoint operators, and especially Schrödinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the one-dimensional case, and in particular on several key examples. The most prominent of these is the Fibonacci Hamiltonian, for which much is known by now and to which an entire section is devoted here. Other examples that are discussed in detail are given by the more general class of Schrödinger operators with Sturmian potentials. We put some emphasis on the methods that have been introduced quite recently in the study of these operators, many of them coming from hyperbolic dynamics. We conclude with a multitude of numerical calculations that illustrate the validity of the known rigorous results and suggest conjectures for further exploration.