Related papers: Aperiodic order and spectral properties
The present notes refer to a lecture delivered on 27 September 2017 in Roscoff during the 2017 Evry Schatzman School. It concerns a general introduction to optical/IR interferometry, including a brief history, a presentation of the basic…
This is an introduction to $p$-adic geometry and $p$-adic analysis focusing on the theme of $p$-adic period mappings. We follow as closely as possible the development of the classical theory of complex period mappings, blending differential…
Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by…
The objective of this paper is to derive the essential invariance and contraction properties for the geometric periodic systems, which can be formulated as a category of differential inclusions, and primarily rendered in the phase…
The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not…
We present a simplified model for dynamical diffraction of particles through a periodic thick perfect crystal based on repeated application of a coherent beam splitting unitary at coarse-grained lattice sites. By demanding translational…
Quasiperiodic magnonic crystals, in contrast to their periodic counterparts, lack strict periodicity which gives rise to complex and localised spin wave spectra characterized by numerous band gaps and fractal features. Despite their…
This is an extended abstract for the Oberwolfach Workshop 2101b \emph{Geometry, Dynamics and Spectrum of Operators on Discrete Spaces} organized by David Damanik, Matthias Keller, Tatiana Smirnova-Nagnibeda, and Felix Pogorzelski. It took…
We describe some basic tools in the spectral theory of Schr\"odinger operator on metric graphs (also known as "quantum graph") by studying in detail some basic examples. The exposition is kept as elementary and accessible as possible. In…
In this paper, we propose a general mechanism for the existence of quasicrystals in spatially extended systems (partial differential equations with Euclidean symmetry). We argue that the existence of quasicrystals with higher order…
In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the…
As a contribution to the inverse scattering problem for classical chaotic systems, we show that one can select sequences of intervals of continuity, each of which yields the information about period, eigenvalue and symmetry of one unstable…
Kirchhoff's scalar diffraction theory is applied throughout photon and electron optics. It is based on the stationary electromagnetic or Schr\"odinger wave equation, and is useful in describing interference phenomena for both light and…
We consider a quasi one-dimensional chain of N chaotic scattering elements with periodic boundary conditions. The classical dynamics of this system is dominated by diffusion. The quantum theory, on the other hand, depends crucially on…
In the recent years, Hopf algebras have been introduced to describe certain combinatorial properties of quantum field theories. I will give a basic introduction to these algebras and review some occurrences in particle physics.
In the setting of modern mathematical logic and model theory, classification theory has been one of the landmark achievements of the field. Likewise, the classification of UHF-algebras and AF-algebras were substantial contributions to the…
Quasicrystals are characterized by quasi-periodic arrangements of atoms. The description of their mechanics involves deformation and a (so called phason) vector field accounting at macroscopic scale of local phase changes, due to atomic…
In a recent Letter we proposed a means to realize a quasicrystal with eight-fold symmetry by trapping particles in an optical potential created by four lasers. The quasicrystals obtained in this way, which are closely related to the…
Modern optical spectrographs and optical interferometers push the limits in the spectral and spatial regime, providing important new tools for the exploration of the universe. In this contribution I outline the complementary nature of…
Quasicrystals are aperiodically ordered solids that exhibit long-range order without translational periodicity, bridging the gap between crystalline and amorphous materials. Due to their lack of translational periodicity, information on…