Strichartz estimate for discrete Schr\"odinger equation on layered King's grid

TL;DR

This paper proves a sharp decay estimate for the discrete Schrödinger equation on the Layered King's Grid, leading to Strichartz estimates, using Newton polyhedra techniques to analyze singularities, and showing faster decay than in 3D lattices.

Contribution

It introduces a novel decay estimate for the discrete Schrödinger equation on LKG, surpassing previous decay rates, and applies Newton polyhedra methods for analysis.

Findings

Established decay rate of .12 for LKG Schrödinger equation

Derived Strichartz estimates from decay results

Demonstrated faster decay than 3D lattice case

Abstract

We establish the sharp \( l^1 \to l^{\infty} \) decay estimate for the discrete Schr\"odinger equation (DS) on the Layered King's Grid (LKG), with a dispersive decay rate of \( \langle t \rangle^{-13/12} \), which is faster than that for $3$-dimensional lattice (\( \langle t \rangle^{-1} \), see \cite{SK05}). This decay estimate enables us to derive the corresponding Strichartz estimate via the standard Keel--Tao argument. Our approach relies on using techniques from Newton polyhedra to analyze singularities.