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This paper is inspired by Wang, Wang and Zhang's work [ Observability andunique continuation inequalities for the Schr\"odinger equation. J. Eur. Math.Soc. 21, 3513--3572 (2019)], where they present several observability andunique continuation inequalities for the free Schr\"{o}dinger equation in$\mathbb{R}^{n}$. We extend all such observability and unique continuationinequalities for the Schr\"{o}dinger equations on half-line with inverse-squarepotentials. Technically, the proofs essentially rely on the representation ofthe solution, a Nazarov type uncertainty principle for the Hankel transform andan interpolation inequality for functions whose Hankel transform have compactsupport.

Observability and unique continuation inequalities for the Schr\"{o}dinger equations with inverse-square potentials