Observable sets for Schr\"odinger equations on combinatorial graphs

TL;DR

This paper investigates the conditions under which sets are observable for Schr"odinger and heat equations on combinatorial graphs, revealing arithmetic obstructions in one dimension and criteria in higher dimensions.

Contribution

It establishes new arithmetic conditions for observability on combinatorial graphs, contrasting with Euclidean cases, and extends results to higher dimensions and discrete tori.

Findings

Observable sets in 1D satisfy a local arithmetic condition.

In higher dimensions, observability holds from the complement of finite sets.

Positive density alone does not guarantee observability on discrete tori.

Abstract

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.