Uncertainty principle for discrete Schrödinger evolution on graphs

Abstract

We consider the Schrödinger evolution on graphs, i.e., solutions to the equation $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, where $\mathcal {A}$ is the set of vertices of the graph and the matrix $(L(\alpha ,\beta ))_{\alpha ,\beta \in \mathcal {A}}$ describes interaction between the vertices, in particular two vertices α and β are connected if $L(\alpha ,\beta )\neq 0$. 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 $u(t,\alpha )$ 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 $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, when $\mathcal {A}$ is a finite set, in terms of the number of the different eigenvalues of the matrix $L(\,\cdot \,,\,\cdot \,)$.