In this article, we prove the decay estimate for the discrete Schrödinger equation (DS) on the hexagonal triangulation. The \(l^1\rightarrow l^\infty \) dispersive decay rate is \(\left\langle t\right\rangle ^{-\frac{3}{4}}\) , which is faster than the decay rate of DS on the 2-dimensional lattice \(\mathbb {Z}^2\) , which is \(\left\langle t\right\rangle ^{-\frac{2}{3}}\) , see [33]. The proof relies on the detailed analysis of singularities of the corresponding phase function and the theory of uniform estimates on oscillatory integrals developed by Karpushkin [15]. Moreover, we prove the Strichartz estimate and give an application to the discrete nonlinear Schrödinger equation (DNLS) on the hexagonal triangulation.
