Weighted estimates for the discrete Laplacian on the cubic lattice

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 $q\ge 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d\ge 3$ with potentials from ${\ell }^p({\mathbb{Z}}^d)$ with suitable $p\ge 1$.

References in corpus (2)

• Scattering on periodic metric graphs
• Branching form of the resolvent at threshold for multi-dimensional discrete Laplacians

Cited by in corpus (3)

• Limiting absorption principle on $L^p$-spaces and scattering theory
• Eigenvalues of periodic difference operators on lattice octant
• Uniform bounds of discrete Birman-Schwinger operators