Scattering of waves by periodic gratings and factorization problems

The paper deals with scattering problems of plane and space waves by periodic gratings in R-2 and R-3, respectively. The plane problem caused a vast physical and mathematical literature starting from the papers of Rayleigh. A new approach to treat this problem is proposed in the present paper. It is based on the possibility to reformulate the scattering problem in terms of abstract ordinary differential equation with constant operator coefficients on a Hilbert space. The solvability of such equations with radiation conditions at the infinity is based on the factorization of the operator symbol of the equation. This approach is general and allows, in particular, to solve the scattering problem of a space wave in R-3.