Application of uncertainty principles for decaying densities to the observability of the Schr\"odinger equation

TL;DR

This paper establishes observability inequalities for the Schrödinger equation in Euclidean space using uncertainty principles tailored to decaying densities, advancing control theory for quantum systems.

Contribution

It introduces new observability inequalities for the Schrödinger equation based on uncertainty principles for decaying densities, extending previous theoretical frameworks.

Findings

Proves observability inequalities for measurable sets with decaying densities.

Utilizes adapted uncertainty principles by Shubin, Vakilian, Wolff, and Kovrijkine.

Enhances understanding of quantum system controllability under decay conditions.

Abstract

In this article, we study the Schr\"odinger equation posed in the Euclidean space. We prove observability inequalities for measurable sets that are thick with respect to decaying densities. The proof relies on quantitative uncertainty principles adapted to decaying densities, notably those established by Shubin, Vakilian, Wolff, and Kovrijkine.