Vector spherical harmonics and scattering integrals

We consider scalar and vectorial non-relativistic bound-free transition form-factors. These integrals are spatial three-dimensional Fourier transforms of the product of the bound and continuum hydrogenlike wavefunctions ϕ nlm ( r ) ϕ K – ( r ), weighted with operators 1, r −1 , r or ∇ . Hamilton gradient operator ∇ r may be applied to either ϕ nlm ( r ) or ϕ K – ( r ). The same type of matrix elements is also calculated by using Slater-type orbitals for ϕ nlm ( r ). Concise general results are obtained for all the form-factors under study in terms of finite quadruple summations. Quantization axis for ϕ nlm ( r ) is held arbitrary and the exact analytical calculations are carried out for any triple nlm of quantum numbers. Vectorial integrals are most elegantly treated by employing circular Cartesian coordinates and the vector spherical harmonics Y jlm ( r ).