Infinite Bar-Joint Frameworks, Crystals and Operator Theory

TL;DR

This paper develops a comprehensive theory of flexibility and rigidity for infinite bar-joint frameworks, introducing operator-theoretic and symbol function methods to analyze infinitesimal flexibility, deformability, and vibrational modes in crystal structures.

Contribution

It introduces a novel operator-theoretic framework and symbol function approach for analyzing rigidity and flexibility in infinite frameworks and crystal models.

Findings

Certain 4-regular planar frameworks are infinitesimally rigid and deformable.

The symbol function determines low-energy vibrational modes in crystal frameworks.

Inversion symmetry leads to RUM surfaces, extending previous results.

Abstract

A theory of flexibility and rigidity is developed for general infinite bar-joint frameworks (G,p). Determinations of nondeformability through vanishing flexibility are obtained as well as sufficient conditions for deformability. Forms of infinitesimal flexibility are defined in terms of the operator theory of the associated infinite rigidity matrix R(G,p). The matricial symbol function of an abstract crystal framework is introduced, being the matrix-valued function on the $d$-torus representing R(G,p) as a Hilbert space operator. The symbol function is related to infinitesimal flexibility, deformability and isostaticity. Various generic abstract crystal frameworks which are in Maxwellian equilibrium, such as certain 4-regular planar frameworks, are proven to be square-summably infinitesimally rigid as well as smoothly deformable in infinitely many ways. The symbol function of a three-dimensional crystal framework determines the infinitesimal wave flexes in models for the low energy vibrational modes (RUMs) in material crystals. For crystal frameworks with inversion symmetry it is shown that the RUMS appear in surfaces, generalising a result of F. Wegner for tetrahedral crystals.