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

Abstract

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 Schrödinger operator $-\Delta +V$ on $L^2(\mathbb{R})$.

References in corpus (1)

• Energy Cost to Make a Hole in the Fermi Sea