High Frequency Resolvent Estimates for Perturbations by Large Long-range Magnetic Potentials and Applications to Dispersive Estimates

We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form ${G=\left(i\nabla+b(x)\right)^2+V(x)}$ on ${L^2({\bf R}^n), n\ge 3}$ , where the magnetic potential b(x) and the electric potential V(x) are long-range and large. As an application, we prove dispersive estimates for the wave group ${{\rm e}^{it\sqrt{G}}}$ in the case n = 3 for potentials b(x), V(x) = O(|x|^?2-δ ) for ${|x|\gg 1}$ , where δ > 0.