On unique continuation for solutions of the Schr{\"o}dinger equation on trees

We prove that if a solution of the time-dependent Schr{\"o}dinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schr{\"o}dinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to a sharp decay at any time. We then use the recent spectral decomposition of the Schr{\"o}dinger operator with compactly supported potential due to Colin de Verdi{\`e}rre and Turc to extend our results in the presence of such potentials. Finally, we use real variable methods first introduced by Escauriaza, Kenig, Ponce and Vega to establish a general sharp result in the case of bounded potentials.