The role of filters and the singular-value decomposition for the inverse Born approximation

The singular-value decomposition (i.e. the generalised inverse) for a linear, compact map is formulated for the noisy inverse scattering problem with general filters and potentials. The differences between the filtered Fourier transform in the variable k and the Born approximation in the variable, tau =k-k', tau =2k sin(¹/₂ theta ), are stressed. The available exact solutions for spherically symmetrical potentials and filters in three dimensions are presented. Finally the solubility of the fixed-energy inverse Born approximation is discussed.