On unique continuation for solutions of the Schrödinger equation on trees

Aingeru Fernandez-Bertolin (1) , Philippe Jaming (1)

1 IMB - Institut de Mathématiques de Bordeaux

We prove that if a solution of the time-dependent Schrödinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schrödinger 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 Schrödinger operator with compactly supported potential due to Colin de Verdièrre 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.

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Variables complexes [math.CV]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Aingeru Fernandez-Bertolin, Philippe Jaming. On unique continuation for solutions of the Schrödinger equation on trees. Annali di Matematica Pura ed Applicata, 2019, ⟨10.1007/s10231-019-00896-z⟩. ⟨hal-01547119v3⟩