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Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts

Tom Potter

Abstract

In this thesis we consider crystal groups in dimension $n$ and their natural unitary representation on $L^2(\mathbb{R}^n)$. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of $L^2(\mathbb{R}^n)$ invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the \textit{central decomposition} guaranteed by direct integral theory.

Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts

Abstract

In this thesis we consider crystal groups in dimension and their natural unitary representation on . We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the \textit{central decomposition} guaranteed by direct integral theory.
Paper Structure (59 sections, 156 theorems, 609 equations, 14 figures, 2 tables)

This paper contains 59 sections, 156 theorems, 609 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Literature Review
  3. Examples
  4. Overview
  5. Crystallographic Groups, Fundamental Domains, and Group Actions
  6. Crystallographic groups
  7. The Point Group and Translation Subgroup
  8. Symmorphic and Nonsymmorphic Crystal Groups
  9. Cross-Sections and Transversals
  10. Fundamental Regions, Domains, and Sets
  11. The Dual Lattice
  12. Table of Relevant Group Actions
  13. The Transversal Omega for the Action of Pi on Parameter Domain
  14. Important Relationships Between Actions
  15. Prerequisite Theory: Unitary Representations and von Neumann Algebras
  16. ...and 44 more sections

Key Result

Theorem 1

The closed, translation-invariant subspaces of $L^2(\mathbb{R}^n)$ are precisely the sets $M_A$. Furthermore, $M_A = M_B$ if and only if the symmetric difference, $A \triangle B$, is a null set.

Figures (14)

  • Figure 1: A pattern illustrating the symmetries of the wallpaper group $pg$.
  • Figure 2: The fundamental set $\Omega_{\mathrm{T}^*}=\left[-\tfrac{1}{2}, \tfrac{1}{2}\right) \times \left[-\tfrac{1}{2}, \tfrac{1}{2}\right)$ for $\mathrm{T}^*$; and the fundamental set $\Omega= \left[ -\tfrac{1}{2}, \tfrac{1}{2}\right) \times \left[-\tfrac{1}{2}, 0 \right]$ for $\Gamma^*$.
  • Figure 3: (a) The set $\Omega_0= \left[ -\tfrac{1}{2}, \tfrac{1}{2}\right) \times \left(-\tfrac{1}{2}, 0 \right)$. (b) The fundamental domain $R = \left( -\tfrac{1}{2}, \tfrac{1}{2}\right) \times \left(-\tfrac{1}{2}, 0 \right)$ for $\Gamma^*$.
  • Figure 4: A pattern illustrating the symmetry of the wallpaper group $p1$; this group has only translation-symmetry.
  • Figure 5: (a) The fundamental set $\Omega=\Omega_{\mathrm{T}^*}=\left[-\tfrac{1}{2}, \tfrac{1}{2}\right) \times \left[-\tfrac{1}{2}, \tfrac{1}{2}\right)$ for $\mathrm{T}^*=\Gamma^*$. We note that $\Omega_0 = \Omega$. (b) The fundamental domain $R = \left( -\tfrac{1}{2}, \tfrac{1}{2}\right) \times \left(-\tfrac{1}{2}, \tfrac{1}{2} \right)$ for $\Gamma^*$.
  • ...and 9 more figures

Theorems & Definitions (438)

  • Theorem
  • Example 1.2.1
  • Example 1.2.2
  • Example 1.2.3
  • Remark 2.1.1
  • Definition 2.1.2
  • Remark 2.2.1
  • Definition 2.2.2
  • Remark 2.2.3
  • Proposition 2.2.4
  • ...and 428 more