  We study the ground state energy and the critical screening parameter of the
Yukawa potential in non-relativistic quantum mechanics. After a short review of
the existing literature on these quantities, we apply fifth-order perturbation
theory to the calculation of the ground state energy, using the exact solutions
of the Coulomb potential together with a cutoff on the principal number
summations. We also perform a variational calculation of the ground state
energy using a Coulomb-like radial wave function and the exact solution of the
corresponding minimization condition. For not too large values of the screening
parameter, close agreement is found between the perturbative and variational
results. For the critical screening parameter, we devise a novel method that
permits us to determine it to ten digits. This is the most precise calculation
of this quantity to date, and allows us to resolve some discrepancies between
previous results.
