Uniform Resolvent Estimates for the Discrete Schrödinger Operator in Higher Dimensions

Abstract

Let $H_0$ be the standard discrete Laplacian on ${\mathbb{Z}}^d$, $d\ge 4$, let $R_0(z)=(H_0-z{)}^{-1}$, and let $p^{\prime }$ denote the Hölder conjugate of $p$, with $p^{\prime }=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from ${\ell }^p({\mathbb{Z}}^d)$ to ${\ell }^{p^{\prime }}({\mathbb{Z}}^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\ge 5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\ge 6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\le p<4/3$ and an ${\ell }^{4/3}$-to-${\ell }^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman--Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.