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

## Analyse Fonctionnelle

Lieu:
Salle Kampé de Fériet M2
Orateur:
Aingeru Fernandez-Bertolin
Affiliation:
Bordeaux
Dates:
Vendredi, 12 Janvier, 2018 - 14:00 - 15:00
Résumé:
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.
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We will show how to use complex analysis and spectral decompositions of Schr\"odinger operators to extend the classical results to homogeneous trees.
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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 $q\ge2$.
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This is joint work with Ph. Jaming (Bordeaux) and L. Vega (UPV/EHU and BCAM)