Uncertainty principle for discrete Schrödinger evolution on graphs

• Issac Álvarez-Romero ^[1]
  1. [1] Norwegian University of Science and Technology

     Norwegian University of Science and Technology

     Noruega

     
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 123, Nº 1, 2018, págs. 51-71
• Idioma: inglés
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• Resumen

  • We consider the Schrödinger evolution on graphs, i.e., solutions to the equation∂tu(t, α)=iβ∈AL(α, β)u(t, β), whereAis the set of vertices of the graph and the matrix(L(α, β))α,β∈Adescribes interaction between the vertices, in particular two verticesαandβare connected ifL(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 a solutionu(t, α)cannot decay too fast along one thread at two differenttimes, unless it vanishes at this thread.We also give a characterization of the dimension of the vector space formed by all the solutionsof∂tu(t, α)=iβ∈AL(α, β)u(t, β), whenAis a finite set, in terms of the number of the different eigenvalues of the matrixL(·,·).