Linear zero mode spectra for quasicrystals

Power, Stephen (2022) Linear zero mode spectra for quasicrystals. Journal of Mathematical Analysis and Applications, 516 (2): 126534. ISSN 0022-247X

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Abstract

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of {linear zero mode spectra} for an aperiodic bar-joint framework $\G$ that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the {limit spectrum} ${\bf L}_{\rm lim}(\G,\ul{a})$ is defined in this way, as a subset of the reciprocal space for a reference basis $\ul{a}$ of the ambient space. A smaller spectrum, the \emph{slippage spectrum} ${\bf L}_{\rm slip}(\G,\ul{a})$, is also defined. For quasicrystal parallelogram frameworks associated with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra coincide and are determined in terms of the geometry of $\G$.