Universal energy functionals for trapped Fermi gases in low dimensions

We study a system of trapped two-component Fermi gases with a zero-range interaction in two dimensions or one dimension. We calculate the one-particle density matrices of these systems at small displacements, from which we show that the N-body energies are linear functionals of the occupation probabilities of single-particle energy eigenstates. A universal energy functional was first derived in 2011 S. Tan, Phys. Rev. Lett. 107, 145302 (2011) for trapped zero-range interacting two-component Fermi gases in three dimensions. We also calculate the asymptotic behaviors of the occupation probabilities of single-particle energy eigenstates at high energies.