[en] A quantum-mechanical theory for the inelastic scattering of slow electrons (ISSE) by surface excitations is developed within the half-space model. The process of transmission of incident electrons into the crystal is described by the homogeneous Schroedinger equation, while the scattering process inside the crystal is described by an inhomogeneous Schroedinger equation. The scattering cross-section for ISSE by surface excitations is derived and is found to be small since it is dependent on an inverse sum of wavevectors which is large. It is also dependent on the fluctuations in the scattering potential. (author)