Relating Diffraction and Spectral Data of Aperiodic Tilings: Towards a Bloch theorem

TL;DR

This paper extends the Bloch theorem to a wide class of aperiodic tilings by establishing a relationship between diffraction patterns and spectral properties via cohomology and K-theory, valid in dimensions up to three.

Contribution

It demonstrates the equivalence of diffraction and spectral data for aperiodic tilings in low dimensions, generalizing the Bloch theorem beyond periodic structures.

Findings

Established conditions for the diffraction-spectral relationship

Proved equivalence of cohomology and K-theory traces in dimensions ≤ 3

Extended Bloch theorem to aperiodic tilings

Abstract

The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by \v{C}ech cohomology of the tiling space) and the spectral properties (of Hamiltonians defined on such tilings) defined by $K$-theory, and to show their equivalence in dimensions $\leq 3$. A theorem makes precise the conditions for this relationship to hold. It can be viewed as an extension of the "Bloch Theorem" to a large class of aperiodic tilings. The idea underlying this result is based on the relationship between cohomology and $K$-theory traces and their equivalence in low dimensions.