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We study the ground-state energy of one-dimensional, non-interacting fermionssubject to an external potential in the thermodynamic limit. To this end, wefix 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 dueto the potential $V$ in the thermodynamic limit $L\to\infty$. We show that thedifference $\mathcal{E}_L(\nu)$ of the two ground-state energies with andwithout potential can be decomposed into a term of order one (leading to theFumi-term) and a term of order $1/L$, which yields the so-called finite sizeenergy. We compute both terms for all possible boundary conditions explicitlyand express them through the scattering data of the one-particle Schr\"odingeroperator $-\Delta +V$ on $L^2(\mathbb R)$.

Ground-state energy of one-dimensional free Fermi gases in the thermodynamic limit