  We consider a Schr\"odinger hamiltonian $H(A,a)$ with scaling critical and
time independent external electromagnetic potential, and assume that the
angular operator $L$ associated to $H$ is positive definite. We prove the
following: 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}\lesssim
t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the
ground level of $L$ and the space dimension $n$. We prove similar results also
for the heat semi-group generated by $H(A,a)$.
