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

TL;DR

This paper analyzes the change in ground-state energy of one-dimensional free Fermi gases in the thermodynamic limit, explicitly computing the leading and finite size correction terms using scattering data.

Contribution

It provides explicit formulas for the energy shift and finite size effects for 1D free Fermi gases with external potentials, based on scattering theory.

Findings

Decomposition of energy difference into Fumi-term and finite size correction

Explicit formulas for both terms in terms of scattering data

Analysis valid for all boundary conditions

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 Schr\"odinger operator $-\Delta +V$ on $L^2(\mathbb R)$.