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We prove a sharp resolvent estimate in scale invariant norms ofAmgon--H\"{o}rmander type for a magnetic Schr\"{o}dinger operator on$\mathbb{R}^{n}$, $n\ge3$\begin{equation*} L=-(\partial+iA)^{2}+V\end{equation*}with large potentials $A,V$ of almost critical decay andregularity.  The estimate is applied to prove sharp smoothing and Strichartz estimates forthe Schr\"{o}dinger, wave and Klein--Gordon flows associated to $L$.

On large potential perturbations of the Schr\"odinger, wave and Klein--Gordon equations