Uncertainty principle for discrete Schrödinger evolution on graphs - Álvarez-Romero, Issac (MATHEMATICA SCANDINAVICA, 2018)
We consider the Schrödinger evolution on graphs, i.e., solutions to the equation $\partial _t u(t,α ) = i\sum _{β \in \mathcal {A}}L(α ,β )u(t,β )$, where $\mathcal {A}$ is the set of vertices of the graph and the matrix $(L(α ,β ))_{α ,β \in \mathcal {A}}$ describes interaction between the vertices, in particular two vertices α and β are connected if $L(α ,β ) eq 0$. 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 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 $\partial _t u(t,α ) = i\sum _{β \in \mathcal {A}}L(α ,β )u(t,β )$, when $\mathcal {A}$ is a finite set, in terms of the number of the different eigenvalues of the matrix $L(\,· \,,\,· \,)$.