Application of uncertainty principles for decaying densities to the observability of the Schrödinger equation

TL;DR

This work addresses observability of the free Schrödinger equation on $\mathbb{R}^d$ when observations are restricted to sets thick with respect to decaying densities. It develops a program that combines Kovrijkhkin’s quantitative uncertainty principle for decaying densities with resolvent-based observability criteria to deduce observability on single-time and two-time-point observation sets. The authors establish three main results: (i) observability for some $T>0$ under $(\gamma,\rho)$-thickness with $\rho(x)=O(|x|^{-1})$; (ii) observability for all $T>0$ when $\rho(x)=O(|x|^{-\alpha})$ with $\alpha>1$ and an explicit $e^{C/T^2}$-type bound; (iii) a two-time-point observability criterion using two decaying densities $\rho_\alpha$ and $\rho_{1/\alpha}$. They also discuss extensions to the fractional Schrödinger equation and to real-valued bounded potentials, linking density-thickness, resolvent control, and observability in a unified framework with potential applications to control and quantum-mechanical monitoring.

Abstract

In this article, we study the Schrödinger 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.

Theorems & Definitions (28)

• Definition 1.1: Observability inequalities
• Definition 1.2: Thick set with respect to a density
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2.1: Kovrizhkin2003
• Remark 2.2
• Theorem 2.3: Mil2005