Calculation of the scattering amplitude from discrete eigenvalues

We show that the scattering amplitude for a target with spherical symmetry can be directly calculated from the discrete eigenvalues of the corresponding stationary wave equation solved under an absorbing boundary condition, and thus uncover a direct connection between the scattering amplitude and the wave number spectrum. As an illustration, we apply this approach to the scattering of an electromagnetic wave by an array of dielectric slabs and find the transmission coefficient of the array to be completely determined by the spacing between adjacent eigenvalues.