Relating diffraction and spectral data of aperiodic tilings: Towards a Bloch theorem

The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by Cech 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 <= 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. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper explores the relationship between the structural aspect of tilings (described by Cech cohomology) and the spectral properties of Hamiltonians defined on those tilings (defined by K-theory), showing their equivalence in dimensions <= 3. A theorem is presented to specify the conditions for this relationship to hold, which can be seen as an extension of the Bloch Theorem to a wide range of aperiodic tilings. The underlying idea behind this result is based on the connection between cohomology and K-theory traces, and their equivalence in low dimensions.