Kinematic Diffraction from a Mathematical Viewpoint

TL;DR

This paper reviews mathematical diffraction theory, focusing on systems with continuous and mixed spectra, and discusses recent advances in understanding their diffraction patterns and inverse structure problems.

Contribution

It provides a comprehensive overview of recent developments in mathematical diffraction theory, especially for systems with stochastic components and mixed spectra.

Findings

Improved understanding of systems with continuous and mixed spectra.

Analysis of homometry and its implications for diffraction.

Discussion of diffraction in stochastic and algebraic dynamical systems.

Abstract

Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Simultaneously, their relevance has grown in practice as well. In this context, the phenomenon of homometry shows various unexpected new facets. This is particularly so for systems with stochastic components. After the introduction to the mathematical tools, we briefly discuss pure point spectra, based on the Poisson summation formula for lattice Dirac combs. This provides an elegant approach to the diffraction formulas of infinite crystals and quasicrystals. We continue by considering classic deterministic examples with singular or absolutely continuous diffraction spectra. In particular, we recall an isospectral family of structures with continuously varying entropy. We close with a summary of more recent results on the diffraction of dynamical systems of algebraic or stochastic origin.