  This work investigates the scattering coefficients for inverse medium
scattering problems. It shows some fundamental properties of the coefficients
such as symmetry and tensorial properties. The relationship between the
scattering coefficients and the far-field pattern is also derived. Furthermore,
the sensitivity of the scattering coefficients with respect to changes in the
permittivity and permeability distributions is investigated. In the linearized
case, explicit formulas for reconstructing permittivity and permeability
distributions from the scattering coefficients is proposed. They relate the
exponentially ill-posed character of the inverse medium scattering problem at a
fixed frequency to the exponential decay of the scattering coefficients.
Moreover, they show the stability of the reconstruction from multifrequency
measurements. This provides a new direction for solving inverse medium
scattering problems.
