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Crystal frameworks, matrix-valued functions and rigidity operators

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Publication date2014
Host publicationConcrete operators, spectral theory, operators in harmonic analysis and approximation: 22nd International Workshop in Operator Theory and its Applications, Sevilla, July 2011
EditorsManuel Cepedello Boiso, Håkan Hedenmalm, Marinus A. Kaashoek, Alfonso Montes Rodríguez, Sergei Treil
Place of PublicationBasel
Number of pages16
ISBN (Electronic)9783034806480
ISBN (Print)9783034806473
<mark>Original language</mark>English

Publication series

NameOperator Theory: Advances and Applications
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878


An introduction and survey is given of some recent work on the infinitesimal dynamics of crystal frameworks, that is, of translationally periodic discrete bond-node structures in ℝd, for d = 2,3,... We discuss the rigidity matrix, a fundamental object from finite bar-joint framework theory, rigidity operators, matrix-function representations and low energy phonons. These phonons in material crystals, such as quartz and zeolites, are known as rigid unit modes, or RUMs, and are associated with the relative motions of rigid units, such as SiO4 tetrahedra in the tetrahedral polyhedral bondnode model for quartz. We also introduce semi-infinite crystal frameworks, bi-crystal frameworks and associated multi-variable Toeplitz operators.