Reconstruction of scattering data by the optical theorem

A method is presented for the optimal reconstruction of the scattering amplitude in the case of missing data. The solution relies on the optical theorem, a consequence of the unitary nature of the scattering matrix, which relates the imaginary part of the forward elastic scattering amplitude to the total cross section. It is shown that the optical theorem leads to a natural constraint on the coefficients of the Fourier expansion of the scattering amplitude. This additional constraint can be used to obtain a regularized solution of the reconstruction problem in the sense of Tikhonov's theory, with a self-adaptive choice of the regularization parameter