Observability and unique continuation inequalities for the Schrödinger equations with inverse-square potentials

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, PR China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 1 March 2025
Accepted: 20 January 2026

Abstract

In this paper, we focus on the Schrödinger equations with inverse-square potentials in dimension one; these special potentials play an important role in the field of mathematical physics. We study several observability and unique continuation inequalities at one time point or at two time points for these equations. These observability and unique continuation inequalities are some new types of quantitative estimates which have appeared in recent literature. Their proofs essentially rely on the representation of the solution, a Nazarov-type uncertainty principle for the Hankel transform, and an interpolation inequality for functions whose Hankel transforms have compact support. Meanwhile, these inequalities can be applied to the controllability of these Schrödinger equations.