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Finite-size energy of non-interacting Fermi gases

We prove the asymptotics of the difference of the ground-state energies of two non-interacting $N$-particle Fermi gases on the half line of length $L$ in the thermodynamic limit up to order $1/L$. We are particularly interested in subdominant terms proportional to $1/L$, called finite-size energy. In the nineties Affleck and co-authors [Aff97, ZA97, AL94] claimed that the finite-size energy equals the decay exponent occuring in Anderson's orthogonality catastrophe. It turns out that the finite-size energy depends on the details of the thermodynamic limit and typically also includes a linear term in the scattering phase shift.