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

## TL;DR

This paper establishes unique continuation properties for solutions of the Schrödinger equation on homogeneous trees, showing that rapid decay at two times implies triviality, and characterizes initial conditions for sharp decay, extending results to bounded potentials.

## Contribution

It introduces new unique continuation results for Schrödinger equations on trees, utilizing spectral theory, complex analysis, and real variable methods, including recent spectral decompositions.

## Key findings

- Solutions with rapid decay at two times are trivial.
- Characterization of initial conditions for sharp decay.
- Extension of results to bounded potentials.

## Abstract

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.