  In this note, we prove the quantitative observability with an explicit
control cost for the 1D Schr\"odinger equation over $\mathbb{R}$ with
real-valued, bounded continuous potential on thick sets. Our proof relies on
different techniques for low-frequency and high-frequency estimates. In
particular, we extend the large time observability result for the 1D free
Schrodinger equation in Theorem 1.1 of Huang-Wang-Wang [20] to any short time.
As another byproduct, we extend the spectral inequality of Lebeau-Moyano [27]
for real-analytic potentials to bounded continuous potentials in the
one-dimensional case.
