Observable sets for the free Schrödinger equation on combinatorial graphs
Zhiqiang Wan, Heng Zhang
TL;DR
This work analyzes observability for the free Schrödinger equation on combinatorial graphs, contrasting discrete lattices with the continuum. It combines Fourier analysis on lattices, discrete uncertainty principles, and TT$^*$ methods to obtain sharp observability results: on $\mathbb{Z}$, thick sets with $\gamma\ge\tfrac12$ are observable at some time while $\gamma<\tfrac12$ can fail; on $\mathbb{Z}^d$, the complement of a finite set is observable at all times; on finite graphs, observability is equivalent to a spectral non-vanishing condition of eigenfunctions on the observation set; and on discrete tori one can construct high-density unobservable sets via product eigenfunctions. The paper also provides a complete arithmetic criterion for Bohr sets and demonstrates explicit unobservable constructions with large density, highlighting arithmetic and spectral mechanisms that govern observability in the discrete setting. Collectively, the results establish discrete analogues and contrasts with continuous observability theory and elucidate the role of sampling, density, and spectral structure in graph-based quantum dynamics.
Abstract
We study observability for the free Schrödinger equation $\partial_t u = iΔu$ on combinatorial graphs $G=(\mathcal{V},\mathcal{E})$. A subset $E\subset\mathcal{V}$ is observable at time $T>0$ if there exists $C(T,E)>0$ such that for all $u_0\in l^2(\mathcal{V})$, $$ \|u_0\|_{l^2(\mathcal{V})}^2 \le C(T,E)\int_0^T \|e^{itΔ}u_0\|_{l^2(E)}^2\,\d t. $$ On the one-dimensional lattice $\mathbb{Z}$, we obtain a sharp threshold for thick sets: if $E\subset\mathbb{Z}$ is $γ$-thick with $γ\geq1/2$, then $E$ is observable at some time; conversely, for every $γ<1/2$ there exists a $γ$-thick set that is not observable at any time. This critical threshold marks the exact point where the discrete lattice departs from the real line: on the lattice it must be attained, whereas on $\R$ any $γ$-thick set with $γ>0$ already suffices. On $\mathbb{Z}^d$ we show that the complements of finite sets are observable at any time $T>0$. This parallels the Euclidean setting $\R^d$: any set that contains the exterior of a finite ball is observable at any time. For finite graphs we give an equivalent characterization of observability in terms of the zero sets of Laplacian eigenfunctions. As an application, we construct unobservable sets of large density on discrete tori, in contrast with the continuous torus $\mathbb{T}^d$, where every nonempty open set is observable.