This paper develops a theory for the sound absorption and scattering of perforated slit absorbers. A rigid plane, perforated periodically in one dimension with absorbing slits, scatters incoming sound waves as discrete wave components in different directions. The absorbing slits are assumed to be line-like in the sense that their width is much shorter than the wavelengths. The equation for the sound field is solved in the wavenumber domain. The slits are described with an impedance description, assuming local reaction of the slits (typically a Helmholtz resonator). The solution is found by means of an inverse transform, back to the spatial domain. This results in an explicit formulation of the sound field, including a sum consisting of components that either radiate energy in discrete directions or are surface waves. A similar sum is also included in a term that can be interpreted as radiation impedance. The explicit expressions for the absorption and scattering coefficients are found with the aid of the radiating part of the scattered and reflected field. Numerical results of the absorption and scattering coefficients are presented. The result is verified with finite element method and compared with the result from an alternative general formulation of the problem.