We consider a Schrodinger 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 $|e^{-itH(A,a)}|
_{L^1to L^infty}lesssim t^{-n/2}$, then $ ||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}|_{L^1to 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 $H(A,a)$.
We study via Carleman estimates the sharpest possible exponential decay for {it waveguide} solutions to the Laplace equation $$(partial^2_t+triangle)u=Vu+Wcdot(partial_t, abla)u,$$ and find a necessary quantitative condition on the exponential decay
in the spatial-variable of nonzero waveguides solutions which depends on the size of $V$ and $W$ at infinity.
We analyze the dispersive properties of a Dirac system perturbed with a magnetic field. We prove a general virial identity; as applications, we obtain smoothing and endpoint Strichartz estimates which are optimal from the decay point of view. We also
prove a Hardy-type inequality for the perturbed Dirac operator.
