Sampling solutions of Schr\"odinger equations on combinatorial graphs

TL;DR

This paper investigates sampling solutions to Schrödinger equations on graphs, identifying conditions under which solutions can be reconstructed from samples on a subset of vertices, with sharp results for bipartite graphs.

Contribution

It introduces a sampling framework for Schrödinger equations on graphs and determines the cut-off frequency for exact reconstruction from samples, especially sharp in bipartite graphs.

Findings

Solutions are determined by samples on a subset of vertices and specific time points.

The cut-off frequency for sampling is explicitly computed.

Results are sharp for bipartite graphs.

Abstract

We consider functions on a graph $G$ whose evolution in time $-\infty<t<\infty$ is governed by a Schr\"{o}dinger type equation with a combinatorial Laplace operator on the right side. For a given subset $S$ of vertices of $G$ we compute a cut-off frequency $\omega>0$ such that solutions to a Cauchy problem with initial data in $PW_{\omega}(G)$ are completely determined by their samples on $S\times \{k\pi/\omega\},$ where $k\in \mathbf{N}$. It is shown that in the case of a bipartite graph our results are sharp.