Simple weight modules over the quantum Schr\"{o}dinger algebra

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