  In this work, we apply a semi-Lagrangian spectral method for the
Vlasov-Poisson system, previously designed for periodic Fourier
discretizations, by implementing Legendre polynomials and Hermite functions in
the approximation of the distribution function with respect to the velocity
variable. We discuss second-order accurate-in-time schemes, obtained by
coupling spectral techniques in the space-velocity domain with a BDF
time-stepping scheme. The resulting method possesses good conservation
properties, which have been assessed by a series of numerical tests conducted
on the standard two-stream instability benchmark problem. In the Hermite case,
we also investigate the numerical behavior in dependence of a scaling parameter
in the Gaussian weight. Confirming previous results from the literature, our
experiments for different representative values of this parameter, indicate
that a proper choice may significantly impact on accuracy, thus suggesting that
suitable strategies should be developed to automatically update the parameter
during the time-advancing procedure.
