Weighted estimates for the discrete Laplacian on the cubic lattice

TL;DR

This paper derives weighted estimates for the discrete Laplacian on cubic lattices, providing new bounds for the associated propagator and resolvent, and applies these to analyze discrete Schrödinger operators with certain potentials.

Contribution

It introduces novel weighted estimates for the discrete Laplacian's propagator and resolvent, extending analysis tools for discrete Schrödinger operators in higher dimensions.

Findings

Established weighted bounds for the group $e^{it riangle}$

Derived resolvent estimates with $ ext{ell}^q$-weights

Applied results to Schrödinger operators with $ ext{ell}^p$ potentials

Abstract

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb Z^d$, and deduce estimates on the group $e^{it\Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q(\mathbb Z^d)$-weights for suitable $q\geq 2$. We apply the obtained results to discrete Schr\"odinger operators in dimension $d\geq 3$ with potentials from $\ell^p(\mathbb Z^d)$ with suitable $p\geq 1$.