Improved time-decay for a class of scaling critical electromagnetic Schrödinger flows

Abstract

We consider a Schrödinger 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 $\Vert e^{-itH(A,a)}{\Vert }_{L^1\to L^{\infty }}\lesssim t^{-n/2}$, then $\Vert |x{|}^{-g(n)}{e}^{-itH(A,a)}|x{|}^{-g(n)}{\Vert }_{L^1\to L^{\infty }}\lesssim t^{-n/2-g(n)}$, $g(n)$ 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 $H(A,a)$.