  We study the entanglement entropies of an interval adjacent to the boundary
of the half line for the free fermionic spinless Schr\"odinger field theory at
finite density and zero temperature, with either Neumann or Dirichlet boundary
conditions. They are finite functions of the dimensionless parameter given by
the product of the Fermi momentum and the length of the interval. The
entanglement entropy displays an oscillatory behaviour, differently from the
case of the interval on the whole line. This behaviour is related to the
Friedel oscillations of the mean particle density on the half line at the
entangling point. We find analytic expressions for the expansions of the
entanglement entropies in the regimes of small and large values of the
dimensionless parameter. They display a remarkable agreement with the curves
obtained numerically. The analysis is extended to a family of free fermionic
Lifshitz models labelled by their integer Lifshitz exponent, whose parity
determines the properties of the entanglement entropies. The cumulants of the
local charge operator and the Schatten norms of the underlying kernels are also
explored.
