Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Critical points of discrete periodic operators

Matthew Faust, Frank Sottile

Abstract

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z^2- and Z^3-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z^2-periodic graphs.

Critical points of discrete periodic operators

Abstract

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z^2- and Z^3-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z^2-periodic graphs.
Paper Structure (18 sections, 25 theorems, 54 equations, 10 figures)

This paper contains 18 sections, 25 theorems, 54 equations, 10 figures.

Table of Contents

  1. Operators on periodic graphs
  2. Operators on periodic graphs
  3. Floquet theory
  4. Critical points of the Bloch variety
  5. Bounding the number of critical points
  6. A little algebraic geometry
  7. Polyhedral bounds
  8. Projective toric varieties
  9. Proof of Theorem B
  10. Facial systems
  11. Facial systems of the critical point equations
  12. Newton polytopes and dense periodic graphs
  13. The Newton polytope of $\Gamma$
  14. Smoothness of the Bloch variety at infinity
  15. Case (v)
  16. ...and 3 more sections

Key Result

Proposition 2.1

A point $(z,\lambda)\in({\mathbb C}^\times)^d\times{\mathbb C}$ is a critical point of the function $\lambda$ on the Bloch variety ${\it Var}(D(z,\lambda))$ if and only if Eq:CPE holds.

Figures (10)

  • Figure 1: A dense periodic graph $\Gamma$ with the convex hull of ${\mathcal{A}}(\Gamma)$.
  • Figure 2: Two ${\mathbb Z}^2$-periodic graphs.
  • Figure 3: A labeling of the hexagonal lattice.
  • Figure 4: A Bloch variety and spectral bands for the hexagonal lattice.
  • Figure 5: Support and Newton polytope of the hexagonal lattice operator.
  • ...and 5 more figures

Theorems & Definitions (50)

  • Example 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Example 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Remark 2.6
  • Proposition 2.7
  • Proposition 2.8
  • ...and 40 more