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Truncated Floquet-Bloch transform for computing the spectral properties of large finite systems of resonators

Habib Ammari, Silvio Barandun, Alexander Uhlmann

Abstract

The truncated Floquet-Bloch transform can be used to characterise the spectral properties of finite periodic and aperiodic large systems of resonators. This paper aims to provide for the first time the mathematical foundations of this transform.

Truncated Floquet-Bloch transform for computing the spectral properties of large finite systems of resonators

Abstract

The truncated Floquet-Bloch transform can be used to characterise the spectral properties of finite periodic and aperiodic large systems of resonators. This paper aims to provide for the first time the mathematical foundations of this transform.

Paper Structure

This paper contains 16 sections, 18 theorems, 77 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Notation and Toeplitz theory
  4. Toeplitz operators and matrices
  5. Delocalisation
  6. Truncated Floquet-Bloch transform
  7. Truncated Floquet-Bloch transform and circulant matrices
  8. Periodic structures
  9. Single resonator or particle with only nearest neighbour interactions
  10. Systems of resonators with long-range interactions
  11. Aperiodic structures
  12. SSH-like structures
  13. Dislocated structures
  14. Compact perturbations
  15. Concluding remarks
  16. ...and 1 more sections

Key Result

Lemma 3.7

Let $C_m(f)$ be a $1$-circulant matrix of size $m$ and $\lambda(e^{\mathbf{i}\alpha})$ its only band. The eigenvalues of $C_m(f)$ are $\lambda(e^{\mathbf{i}\alpha_j})$ and the eigenvectors are given by where $\alpha_j \coloneqq 2\pi\frac{j}{m}$ for $0\leq j\leq m-1$ and the superscript $\top$ denotes the transpose.

Figures (8)

  • Figure 1: Illustration of different steps behind the truncated Floquet-Bloch transform in reconstructing the quasiperiodicity of an eigenmode. The map $\mathcal{F}$ is the discrete Fourier transform and $\Phi$ is defined in \ref{['eq: phi map slicing']}.
  • Figure 2: The "original" truncated Floquet-Bloch transform presented in ammari.davies.ea2023Convergenceammari.davies.ea2023Spectral in blue and the version from \ref{['def: tfbt']} with indices scaled as $r\mapsto 2\pi\frac{r}{m}$ for $0\leq r\leq m$ in red. The plot shows the two techniques applied to an eigenvector of a circulant matrix of size $20\times 20$.
  • Figure 3: Band structure reconstruction for systems with only nearest neighbour interactions (eigenpairs of $\mathcal{C}_m$ from \ref{['eq: form cap mat 1D']}). The solid lines show the traces of the symbol's eigenvalues (i.e., the actual periodic bands) while the green circles show the reconstructed discrete bands.
  • Figure 4: Band reconstruction for a three-dimensional dimer system composed of 100 resonators. Fourier coefficient of the matrix of interest decay slowly with approximately $\mathcal{O}(n^{-1.2})$. Nevertheless, the band structure is perfectly reconstructed.
  • Figure 5: Band reconstruction for $T_m(f)$ with $f$ as in \ref{['eq: exp decaying symbol']} with exponentially decaying Fourier coefficients.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Remark 3.1
  • Definition 3.2: Band functions
  • Definition 3.4: Banded matrices
  • Definition 3.5: $r$-banded approximation
  • Definition 3.6: Circulant matrix
  • Lemma 3.7
  • Definition 3.8: Discrete Brillouin zone
  • Definition 3.9: Quasiperiodic extension
  • Proposition 3.10
  • Definition 3.11: Delocalisation
  • ...and 24 more