$L^p$ estimates for discrete Schrödinger equations on $\mathbb Z^d$
Studia Mathematica
MSC: Primary 39A12; Secondary 35R02
DOI: 10.4064/sm250421-13-10
Published online: 11 May 2026
Abstract
Let $\varDelta_d$ defined be the discrete Laplacian on $\mathbb Z^d$, $d\ge 1$, defined by $$ \varDelta_d f(x) = \sum_{j=1}^d -[f(x+e_j)+f(x-e_j)-2f(x)], \ \quad x\in \mathbb Z^d, $$ where $\{e_j: j=1, \ldots , d\}$ is the standard basis for $\mathbb R^d$.
The main aim of this paper is to prove the $\ell ^p(\mathbb Z^d)$ estimates for the solutions to the Schrödinger equation $$\begin{cases} i \dfrac{\partial u}{\partial t} + \varDelta_d u =0,\\ u (x,0) = f(x). \end{cases} $$ Our approach is inspired by harmonic analysis methods such as the estimates of joint spectral multipliers and the theory of Hardy spaces associated with the discrete Laplacian.
