Stability estimate for an inverse problem for the time harmonic magnetic Schr\"odinger operator from the near and far field pattern

We derive conditional stability estimates for inverse scattering problems related to time harmonic magnetic Schr"odinger equation. We prove logarithmic type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near field or far field maps. Our approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions.