  In the present paper, using the technique of localization, we determine the
center of the quantum Schr\"{o}dinger algebra $\S_q$ and classify simple
modules with finite-dimensional weight spaces over $\S_q$, when $q$ is not a
root of unity. It turns out that there are four classes of such modules: dense
$U_q(\mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules,
and twisted modules of highest weight modules.
