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Crystal Frameworks, Matrix-valued Functions and Rigidity Operators

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Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 236))

Abstract

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.

Mathematics Subject Classification (2010). Primary 52C75; Secondary 46T20.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK

    S. C. Power

Authors
  1. S. C. Power
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Correspondence to S. C. Power .

Editor information

Editors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Manuel Cepedello Boiso

  2. Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden

    Håkan Hedenmalm

  3. Department of Mathematics & Computer Sci, Free University Amsterdam, Amsterdam, The Netherlands

    Marinus A. Kaashoek

  4. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Alfonso Montes Rodríguez

  5. Department of Mathematics, Brown University, Providence, Rhode Island, USA

    Sergei Treil

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Power, S.C. (2014). Crystal Frameworks, Matrix-valued Functions and Rigidity Operators. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_26

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