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

This paper is inspired by Wang, Wang and Zhang’s work [ Observability and unique continuation inequalities for the Schrödinger equation. J. Eur. Math. Soc. 21, 3513–3572 (2019)], where they present several observability and unique continuation inequalities for the free Schrödinger equation in ${\mathbb{R}}^n$. We extend all such observability and unique continuation inequalities for the Schrödinger equations on half-line with inverse-square potentials. Technically, the 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 transform have compact support.