Uncertainty principle for discrete Schr\"odinger evolution on graphs

TL;DR

This paper establishes an uncertainty principle for Schrödinger evolution on graphs with a web-like structure, showing solutions cannot decay rapidly along a thread at two times unless they are trivial, and characterizes solution spaces for finite graphs.

Contribution

It introduces a novel uncertainty principle for discrete Schrödinger equations on complex graph structures and characterizes the solution space dimension based on eigenvalues.

Findings

Solutions cannot decay too fast along a thread at two different times unless they vanish.

Characterization of the solution space dimension in finite graphs based on eigenvalues.

Provides insights into the behavior of quantum evolution on graph structures.

Abstract

We consider the Schr\"odinger evolution on graph, i.e. solution to the equation $\partial_tu(t,\alpha)=i\sum_{\beta\in\mathcal{A}}L(\alpha,\beta)u(t,\beta)$, here $\mathcal{A}$ is the set of vertices of the graph and the matrix $(L(\alpha,\beta))_{\alpha,\beta\in\mathcal{A}}$ describes interaction between the vertices, in particular two vertices $\alpha$ and $\beta$ are connected if $L(\alpha,\beta)\neq0$. We assume that the graph has a "web-like" structure, i.e, it consists of an inner part, formed by a finite number of vertices, and some threads attach to it. We prove that such solution $u(t,\alpha)$ cannot decay too fast along one thread at two different times, unless it vanishes at this thread. We also give a characterization of the dimension of the vector space formed by all the solutions of $\partial_tu(t,\alpha)=i\sum_{\beta\in\mathcal{A}}L(\alpha,\beta)u(t,\beta)$ when $\mathcal{A}$ is a finite set, in terms of the number of the different eigenvalues of the matrix $L(\cdot,\cdot)$