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We consider a Schr\"odinger hamiltonian $H(A,a)$ with scaling critical andtime independent external electromagnetic potential, and assume that theangular operator $L$ associated to $H$ is positive definite. We prove thefollowing: if $\|e^{-itH(A,a)}\|_{L^1\to L^\infty}\lesssim t^{-n/2}$, then  $ \||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}\|_{L^1\to L^\infty}\lesssimt^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on theground level of $L$ and the space dimension $n$. We prove similar results alsofor the heat semi-group generated by $H(A,a)$.

Improved time-decay for a class of scaling critical electromagnetic Schr\"odinger flows