  We study the entanglement entropies of an interval on the infinite line in
the free fermionic spinless Schr\"odinger field theory at finite density and
zero temperature, which is a non-relativistic model with Lifshitz exponent
$z=2$. We prove that the entanglement entropies are finite functions of one
dimensionless parameter proportional to the area of a rectangular region in the
phase space determined by the Fermi momentum and the length of the interval.
The entanglement entropy is a monotonically increasing function. By employing
the properties of the prolate spheroidal wave functions of order zero or the
asymptotic expansions of the tau function of the sine kernel, we find analytic
expressions for the expansions of the entanglement entropies in the asymptotic
regimes of small and large area of the rectangular region in the phase space.
These expansions lead to prove that the analogue of the relativistic entropic
$C$ function is not monotonous. Extending our analyses to a class of free
fermionic Lifshitz models labelled by their integer dynamical exponent $z$, we
find that the parity of this exponent determines the properties of the
bipartite entanglement for an interval on the line.
