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

We consider a Schrödinger hamiltonian H(A,a)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)‖L1→L∞≲t−n/2‖e−itH(A,a)‖L1→L∞≲t−n/2, then ‖|x|−g(n)e−itH(A,a)|x|−g(n)‖L1→L∞≲t−n/2−g(n)‖|x|−g(n)e−itH(A,a)|x|−g(n)‖L1→L∞≲t−n/2−g(n), 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)H(A,a).