An Eulerian drift-kinetic Vlasov code is used to study the interaction of lower hybrid (LH) waves with a plasma slab. We investigate the evolution of the LH spectrum, as well as the formation and evolution of the tail of the distribution function, as the LH waves propagate and damp across the plasma slab. Results obtained with single and compound spectra of LH waves are presented. In the case of compound spectra, the synergy between the fast and the slow spectrum of lower hybrid waves is studied. Test particle diagnostics is applied to estimate the velocity diffusion and the LH-driven spatial diffusion across the magnetic field. The Vlasov code gives an accurate representation of the distribution function in the low density region of the phase-space. The absence of noise in the Vlasov code allows for accurate calculations of the diffusion coefficients.