Discrete Four-Order Schrödinger Equation on the Hexagonal Triangulation

Abstract

In this paper, we establish a decay estimate for the discrete four-order Schrödinger equation on the hexagonal triangulation with $γ$=0. The proof is based on the uniform estimates of oscillatory integrals, as developed by Karpushkin, along with a key result by Varchenko. Our result is to show the $l^1$→$l^∞$ dispersive decay rate is $⟨t⟩^{−σ}$ for any $0<\sigma<\frac{1}{2}.$ Additionally, we provide estimates for the inhomogeneous discrete fourth-order Schrödinger equation with $γ$=0.