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arXiv · 2007.15961

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

Eric Akkermans·Yaroslav Don·Jonathan Rosenberg·Claude L. Schochet
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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.

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Eric Akkermans, Yaroslav Don, Jonathan Rosenberg, Claude L. Schochet. 2020-07-31. Relating Diffraction and Spectral Data of Aperiodic Tilings: Towards a Bloch theorem. https://doi.org/10.1016/j.geomphys.2021.104217

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