Observable sets for Schr\"odinger equations on combinatorial graphs

We study observable sets for Schr\"odinger equations on combinatorial graphs. For one-dimensional lattice Schr\"odinger operators $H=-\Delta_{\mathrm{disc}}+V$ with $V(n)\to c\in\mathbb R$ as $|n|\to\infty$, we prove that a set $E\subset\mathbb Z$ is observable at some time, equivalently at any time, if and only if it satisfies a local arithmetic condition. This reveals an arithmetic obstruction absent from the Euclidean theory, where thickness is the decisive condition. The same criterion also characterizes observability for the corresponding heat equation on $\mathbb Z$. In higher-dimensional lattices, we prove observability from the complement of any finite set. We further obtain arithmetic criteria on discrete tori, showing that positive density alone does not ensure observability.