  This proposal relates to the design, analysis and application of a novel
numerical scheme for the solution of axisymmetric scattering problems. To this
end, a procedure is introduced to iteratively evaluate the solution of the
Lippmann-Schwinger integral equation in $O(N\log^2 N)$ operations, where $N$ is
the number of the discretization points. The method achieves its efficiency
through the use of the addition theorem and Fast Legendre Transforms (FLT).
  For globally smooth sound velocities/refractive indexes the method is
spectrally accurate. More generally the order of convergence is tied to and in
fact, limited by, the smoothness of the solution.
