# Spectral Properties of Quasicrystals via Analysis, Dynamics, and Geometric Measure Theory (15w5083)

Arriving in Oaxaca, Mexico Sunday, September 27 and departing Friday October 2, 2015

## Objectives

The Fibonacci sequence is the central one-dimensional quasi-crystal model. It arises in a suitably chosen cut-and-project formalism, where both the physical space and the internal space are one-dimensional. An additional feature of the Fibonacci sequence is its invariance under a substitution rule. This invariance gives rise to a powerful tool that may be employed in the spectral analysis of the Hamiltonian associated with the Fibonacci sequence, the so-called trace map. As a consequence, much of the spectral analysis reduces to a study of a simple dynamical system. While this connection was realized early on, it was only recently that this connection has been exploited to the fullest extent possible, making connections with many recent results in hyperbolic dynamics, and this has led to a multitude of very detailed results for this special case. These developments are still very much underway and are expected to lead to many more interesting applications in mathematical physics. For example, further recent applications concern the Ising model and quantum walks in Fibonacci environments. Thus, the first objective of the proposed half-workshop is to present and discuss all the recent developments for Fibonacci-based models in mathematical physics and to stimulate further research in this direction.

Other one-dimensional quasi-crystal models may not be invariant under a substitution rule, but the standard ones (e.g., the Sturmian sequences) are at least S-adic. That is, while they are not generated by a single substitution, they are generated by a specific sequence of substitutions. The corresponding dynamical problem is then to study the iterates of a point under a sequence of corresponding maps. The basic hyperbolic theory for this setup is still in its infancy and would have to be developed in order to make similarly strong advances for Sturmian models. Of course, once such a theory has been developed, it will have many other potential applications. A discussion centered around these issues is the second objective of the proposed half-workshop.

As mentioned before, very few rigorous results exist for higher-dimensional quasicrystal models. In particular, nothing is known about the shape of the spectrum, the spectral type, and quantum transport properties. This is in part due to the fact that the general theory of Schrödinger operators is less extensively developed in dimensions greater than one. Another reason is that some key observations have yet to be made. For example, the Penrose tiling is the natural two-dimensional analog of the Fibonacci sequence and shares many crucial features with it. In particular, it can be obtained both as a cut-and-project set and by way of an iterated substitution procedure. The latter is what leads to the trace map in the Fibonacci case and thereby allows one to perform a dynamical study with spectral implications. This general approach is not limited to one dimension. In fact, we feel that the fact that some of the Penrose tilings can be constructed via a substitution scheme can be used to provide some kind of description of the spectral properties of Laplacian on Penrose tilings. Different ways to use this approach where suggested by Teplyaev (via the spectral decimation method) and by Bellissard. A discussion of these issues will be the third objective.

There are simple higher-dimensional models which are obtained by taking Cartesian products of one-dimensional models. These models have been proposed and studied numerically by physicists (e.g., Sire, Lifshitz, Even-Dar Mandel) and recently some of their observations and predictions could be established rigorously. As a consequence of the product structure of the model, the spectrum of the product model is the set-wise sum (or product, depending on the setup) of the spectra of the one-dimensional components. Since the latter have Cantor spectra, one is led to an investigation of the sum (or product) of Cantor subsets of the real line. Moreover, there is a canonical probability measure supported on the spectrum, the density of states measure, which results as the thermodynamic limit of the eigenvalue distribution of finite volume restrictions of the operator. The density of states measure of the product model turns out the be the convolution of the density of states measures of the one-dimensional factor operators. Thus, one is led to a study of convolutions of singular probability measure on the real line. Questions of this kind have been studied for a long time by researchers in fractal geometry, geometric measure theory, and dynamics and many results are known for special cases. The application of these results to the models at hand is not straightforward. Nevertheless, this connection has proved to be very fruitful and has produced results for products of the one-dimensional Fibonacci model with itself. There is clearly much more to do in this direction and there are natural current further developments. This will constitute the fourth objective of this meeting.

Other one-dimensional quasi-crystal models may not be invariant under a substitution rule, but the standard ones (e.g., the Sturmian sequences) are at least S-adic. That is, while they are not generated by a single substitution, they are generated by a specific sequence of substitutions. The corresponding dynamical problem is then to study the iterates of a point under a sequence of corresponding maps. The basic hyperbolic theory for this setup is still in its infancy and would have to be developed in order to make similarly strong advances for Sturmian models. Of course, once such a theory has been developed, it will have many other potential applications. A discussion centered around these issues is the second objective of the proposed half-workshop.

As mentioned before, very few rigorous results exist for higher-dimensional quasicrystal models. In particular, nothing is known about the shape of the spectrum, the spectral type, and quantum transport properties. This is in part due to the fact that the general theory of Schrödinger operators is less extensively developed in dimensions greater than one. Another reason is that some key observations have yet to be made. For example, the Penrose tiling is the natural two-dimensional analog of the Fibonacci sequence and shares many crucial features with it. In particular, it can be obtained both as a cut-and-project set and by way of an iterated substitution procedure. The latter is what leads to the trace map in the Fibonacci case and thereby allows one to perform a dynamical study with spectral implications. This general approach is not limited to one dimension. In fact, we feel that the fact that some of the Penrose tilings can be constructed via a substitution scheme can be used to provide some kind of description of the spectral properties of Laplacian on Penrose tilings. Different ways to use this approach where suggested by Teplyaev (via the spectral decimation method) and by Bellissard. A discussion of these issues will be the third objective.

There are simple higher-dimensional models which are obtained by taking Cartesian products of one-dimensional models. These models have been proposed and studied numerically by physicists (e.g., Sire, Lifshitz, Even-Dar Mandel) and recently some of their observations and predictions could be established rigorously. As a consequence of the product structure of the model, the spectrum of the product model is the set-wise sum (or product, depending on the setup) of the spectra of the one-dimensional components. Since the latter have Cantor spectra, one is led to an investigation of the sum (or product) of Cantor subsets of the real line. Moreover, there is a canonical probability measure supported on the spectrum, the density of states measure, which results as the thermodynamic limit of the eigenvalue distribution of finite volume restrictions of the operator. The density of states measure of the product model turns out the be the convolution of the density of states measures of the one-dimensional factor operators. Thus, one is led to a study of convolutions of singular probability measure on the real line. Questions of this kind have been studied for a long time by researchers in fractal geometry, geometric measure theory, and dynamics and many results are known for special cases. The application of these results to the models at hand is not straightforward. Nevertheless, this connection has proved to be very fruitful and has produced results for products of the one-dimensional Fibonacci model with itself. There is clearly much more to do in this direction and there are natural current further developments. This will constitute the fourth objective of this meeting.