  In this paper, we generalize the $A_\fz$ extrapolation theorem in \cite{cmp}
and the $A_p$ extrapolation theorem of Rubio de Francia to Schr\"odinger
settings. In addition, we also establish the weighted vector-valued
inequalities for Schr\"odinger type maximal operators by using weights
belonging to $ A_p^{\rho,\tz}$ which includes $A_p$. As their applications, we
establish the weighted vector-valued inequalities for some Sch\"odinger type
operators and pseudo-differential operators.
