Periodic approximations of Schrödinger operators associated with quasicrystals

• Applicant: Dr. Siegfried Beckus
• Subject Area: Mathematics
• Term: since 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 412141125

A Schrödinger operator associated with a solid describes the quantum motion of an electron in the material. The examination of solids with a long range aperiodic order has soared up since their discovery by Shechtman in 1984 for which he was awarded the Nobel Prize in Chemistry. Such materials are called quasicrystals. The spectral study of the associated Schrödinger operators is a challenging task that has mainly been solved for one-dimensional systems. On the other hand, a general technique is missing to treat such operators in higher dimensions. An established method for mathematicians and physicists is to use periodic approximations of the operators since their spectral properties can be examined with Floquet-Bloch theory.With my coauthors, I have developed a new approach where approximations of the operators are defined by approximating the underlying dynamical system. We showed that the convergence of the dynamical systems implies the convergence of the spectra of the associated Schrödinger operators and the convergence of measured quantities like the density of states measure and the autocorrelation measure. In addition, we have established estimates for the rate of convergence of the spectra for the class of symbolic dynamical systems. All these results are valid in higher dimensions and provide a new approach to investigate Schrödinger operators associated with higher dimensional quasicrystals. Within the present proposal, I seek to significantly develop this theory. One of the main tasks is the construction of periodic dynamical systems that converge to the dynamical system of a quasicrystal in higher dimensions.The first part focuses on a one-dimensional model, called Kohmoto model, which is an important class of quasicrystals. I seek to use our new approach to provide a deeper insight in the dynamical defects that can be created by approximating the operators and to estimate the rate of the convergence of the spectra explicitly. In the second part, I plan to extend the qualitative estimates of the spectra, which we obtained for specific higher dimensional solids, to the class of Delone sets. This is of particular interest since examples such as the Penrose tiling and the octagonal lattice, which are relevant for physicists, are described by Delone sets. Furthermore, such explicit estimates are crucial for numerical results and for controlling the approximations analytically. In the third project, I seek to find sufficient conditions and explicit constructions for periodic approximations of dynamical systems associated with Delone sets representing quasicrystals. A special focus is put on the class of primitive substitutions and cut-and-project sets.

• DFG Programme: Research Grants