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On the dispersive estimates for the discrete Schrödinger equation on a honeycomb lattice

Younghun Hong, Yukihide Tadano, Changhun Yang

TL;DR

This work analyzes the dispersive properties of the discrete Schrödinger equation on a honeycomb lattice, modeling graphene via tight-binding. It identifies degenerate frequencies along three symmetric curves that meet at Dirac points and derives frequency-localized $L^1\to L^\infty$ decay and Strichartz estimates for the linear flow, with decay rates $O(|t|^{-2/3})$ away from Dirac points and $O(|t|^{-5/6})$ near them; the authors also treat intersections of degenerate curves, obtaining $O(|t|^{-2/3})$ decays there. A direct, elementary approach to oscillatory integrals with degenerate phases is developed, enabling rigorous nonlinear scattering results for a discrete NLS on the honeycomb lattice. The combination of explicit phase/Hessian analysis and geometric decomposition of the frequency domain provides a complete dispersive framework for graphene-like tight-binding models and informs nonlinear dynamics on such lattices. Overall, the paper advances understanding of wave propagation in graphene-inspired lattices by delivering sharp, frequency-localized dispersive bounds and nonlinear applications grounded in elementary methods.

Abstract

The discrete Schrödinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate frequencies of the dispersion relation are completely characterized by three symmetric periodic curves (Theorem 2.1), and that the three curves meet at Dirac points where conical singularities appear (see Figure 2.1). Based on this observation, we prove the $L^1\to L^\infty$ dispersion estimates for the linear flow depending on the frequency localization (Theorem 2.3). Collecting all, we obtain the dispersion estimate with $O(|t|^{-2/3})$ decay as well as Strichartz estimates. As an application, we prove small data scattering for a nonlinear model (Theorem 2.10). The proof of the key dispersion estimates is based on the associated oscillatory integral estimates with degenerate phases and conical singularities at Dirac points. Our proof is direct and uses only elementary methods.

On the dispersive estimates for the discrete Schrödinger equation on a honeycomb lattice

TL;DR

This work analyzes the dispersive properties of the discrete Schrödinger equation on a honeycomb lattice, modeling graphene via tight-binding. It identifies degenerate frequencies along three symmetric curves that meet at Dirac points and derives frequency-localized decay and Strichartz estimates for the linear flow, with decay rates away from Dirac points and near them; the authors also treat intersections of degenerate curves, obtaining decays there. A direct, elementary approach to oscillatory integrals with degenerate phases is developed, enabling rigorous nonlinear scattering results for a discrete NLS on the honeycomb lattice. The combination of explicit phase/Hessian analysis and geometric decomposition of the frequency domain provides a complete dispersive framework for graphene-like tight-binding models and informs nonlinear dynamics on such lattices. Overall, the paper advances understanding of wave propagation in graphene-inspired lattices by delivering sharp, frequency-localized dispersive bounds and nonlinear applications grounded in elementary methods.

Abstract

The discrete Schrödinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate frequencies of the dispersion relation are completely characterized by three symmetric periodic curves (Theorem 2.1), and that the three curves meet at Dirac points where conical singularities appear (see Figure 2.1). Based on this observation, we prove the dispersion estimates for the linear flow depending on the frequency localization (Theorem 2.3). Collecting all, we obtain the dispersion estimate with decay as well as Strichartz estimates. As an application, we prove small data scattering for a nonlinear model (Theorem 2.10). The proof of the key dispersion estimates is based on the associated oscillatory integral estimates with degenerate phases and conical singularities at Dirac points. Our proof is direct and uses only elementary methods.

Paper Structure

This paper contains 26 sections, 23 theorems, 222 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Tight-binding model for graphene
  3. Main results
  4. Characterization of degenerate frequencies
  5. Dispersion and Strichartz estimates
  6. Nonlinear application
  7. Notations
  8. Organization of the paper
  9. Acknowledgment
  10. Characterization of degenerate frequencies: Proof of Theorem \ref{['thm: gradient and hessian']}
  11. Reduction to the oscillatory integral estimate
  12. Oscillatory integral localized at a Dirac point
  13. Reduction to the degenerate oscillatory integral estimate
  14. Preliminary degenerate oscillatory integral estimate
  15. Refined degenerate oscillatory integral estimate
  16. ...and 11 more sections

Key Result

Theorem 2.1

For any frequency $\mathbf{k}\in \mathbb{R}^2/\mathbf{\Lambda}^*$ such that $\varphi(\mathbf{k})\neq0$, i.e, $\mathbf{k}$ is not a Dirac point, we have where $\mathbf{V}=[\mathbf{v}_1\ \mathbf{v}_2]$, and As a consequence, $\textup{det}(\nabla^2\varphi)(\mathbf{k})=0$ if and only if

Figures (3)

  • Figure 1.1: Periodic structure of the honeycomb lattice $\mathbf{H}$
  • Figure 2.1: Degenerate frequency curves.
  • Figure 5.1: Frequency domain for $\mathbb{C}_{\mathbf{k}\approx\mathbf{K}_\star}(t; v, \theta)$

Theorems & Definitions (55)

  • Definition 1.1: Fourier transform on the sub-lattice $\mathbf{\Lambda}$
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1: Characterization of degenerate frequencies
  • Remark 2.2
  • Theorem 2.3: Frequency localized dispersion estimates
  • Remark 2.4: Decay rate
  • Remark 2.5: Non-degenerate frequency case $j=0$
  • Remark 2.6: Degenerate frequency case $j=1,2,3$; direct proof
  • Corollary 2.7: Dispersion estimate
  • ...and 45 more