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We study the action of metaplectic operators on Wiener amalgam spaces, givingupper bounds for their norms. As an application, we obtain new fixed-timeestimates in these spaces for Schr\"odinger equations with general quadraticHamiltonians and Strichartz estimates for the Schr\"odinger equation withpotentials $V(x)=\pm|x|^2$.

Metaplectic representation on Wiener amalgam spaces and applications to the Schr\"odinger equation