Uncertainty principle for discrete Schr\"odinger evolution on graphs
Abstract
We consider the Schr\"odinger evolution on graph, i.e. solution to the equation ∂tu(t,α)=iΣβ∈AL(α,β)u(t,β), here A is the set of vertices of the graph and the matrix (L(α,β))α,β∈A describes interaction between the vertices, in particular two vertices α and β are connected if L(α,β)≠0. 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,α) 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 ∂tu(t,α)=iΣβ∈AL(α,β)u(t,β) when A is a finite set, in terms of the number of the different eigenvalues of the matrix L(·,·)
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