Unique continuation inequalities for the Dunkl-Schr\"odinger equation via uncertainty principles

TL;DR

This paper proves unique continuation inequalities for the Dunkl--Schr"odinger equation using uncertainty principles related to the Dunkl transform, demonstrating strong annihilating pairs and quantitative properties.

Contribution

It introduces new unique continuation inequalities for the Dunkl--Schr"odinger equation based on uncertainty principles and annihilating pairs for the Dunkl transform.

Findings

Pairs of (,)-thin sets form strong annihilating pairs for the Dunkl transform

Quantitative uncertainty principles are established for the Dunkl transform

Unique continuation properties for solutions to the Dunkl--Schr"odinger equation are derived

Abstract

In this paper, we establish unique continuation inequalities at two time points for the Dunkl--Schr\"odinger equation. The proof is based on quantitative uncertainty principles for the Dunkl transform. In particular, we prove that pairs of (\varepsilon,k)-thin sets form strong annihilating pairs for the Dunkl transform, which yields quantitative unique continuation properties for solutions to the Dunkl--Schr\"odinger equation.