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.
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