Weighted estimates for the Laplacian on the cubic lattice

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb{Z}^d$, and deduce estimates on the group $e^{i t \Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q (\mathbb{Z}^d)$-weights for suitable $q \geqslant 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d \geqslant 3$ with potentials from $\ell^p (\mathbb{Z}^d)$ with suitable $p \geqslant1$.

Keywords

discrete Laplacian, resolvent, Bessel function, Birman–Schwinger

2010 Mathematics Subject Classification

33C10, 47A40, 81Q10, 81Q35

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The authors were supported by the RSF grant No 18-11-00032, and by the Danish Council for Independent Research grant No 1323-00360.

Received 6 November 2018

Accepted 7 May 2019

Published 7 October 2019