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

Abstract: The discrete Schr\"odinger 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 1.1). 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.