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We prove the following: if $\\|e^{-itH(A,a)}\\|_{L^1\\to L^\\infty}\\lesssim t^{-n/2}$, then\n  $ \\||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}\\|_{L^1\\to L^\\infty}\\lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. 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