  We study the ground-state energy of one-dimensional, non-interacting fermions
subject to an external potential in the thermodynamic limit. To this end, we
fix some (Fermi) energy $\nu>0$, confine fermions with total energy below $\nu$
inside the interval $[-L,L]$ and study the shift of the ground-state energy due
to the potential $V$ in the thermodynamic limit $L\to\infty$. We show that the
difference $\mathcal{E}_L(\nu)$ of the two ground-state energies with and
without potential can be decomposed into a term of order one (leading to the
Fumi-term) and a term of order $1/L$, which yields the so-called finite size
energy. We compute both terms for all possible boundary conditions explicitly
and express them through the scattering data of the one-particle Schr\"odinger
operator $-\Delta +V$ on $L^2(\mathbb R)$.
