Unique Continuation for the Schr\"odinger equation on homogeneous trees

Analyse Fonctionnelle

Salle Kampé de Fériet M2
Aingeru Fernandez-Bertolin
Vendredi, 12 Janvier, 2018 - 14:00 - 15:00
In this talk, we will see that if a solution of the time-dependent Schr\"odinger equation
\partial_tu=\Delta u+Vu
on a homogeneous tree (with $\Delta$ the graph Laplacian) decays fast enough at two distinct times then the solution is trivial. This can be understood as a dynamic version of the Hardy Uncertainty Principle on homogeneous trees, a classical result in Harmonic Analysis proved via complex analysis.
We will show how to use complex analysis and spectral decompositions of Schr\"odinger operators to extend the classical results to homogeneous trees.
Since $\mathbb{Z}$ can be understood as a homogeneous tree of degree 1, we will focus on this setting to show also a proof based on real calculus, based on the results of Escauriaza, Kenig, Ponce and Vega in $\mathbb{R}^d$, and then we will extend the proof to homogeneous trees of degree $q\ge2$.
This is joint work with Ph. Jaming (Bordeaux) and L. Vega (UPV/EHU and BCAM)