$L^p$ resolvent estimates for magnetic Schr\"odinger operators with unbounded background fields

We prove $L^p$ and smoothing estimates for the resolvent of magnetic Schr"odinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schr"odinger and Pauli operators with complex electromagnetic potentials.