Uncertainty principle for discrete Schrödinger evolution on graphs
We consider the Schrödinger evolution on graphs, i.e., solutions to the equation _t u(t,α ) = i_β A L(α ,β )u(t,β ), where 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 ) 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 _t u(t,α ) = i_β A L(α ,β )u(t,β ), when A is a finite set, in terms of the number of the different eigenvalues of the matrix L( · · ). , number=1, journal=MATHEMATICA SCANDINAVICA, author=Álvarez-Romero, Issac, year=2018, month=Aug., pages=51–71