Statistics of fermions in a $d$-dimensional box near a hard wall

Understand

We study $N$ noninteracting fermions in a domain bounded by a hard wall potential in $d \geq 1$ dimensions.

• We show that for large $N$, the correlations at the edge of the Fermi gas (near the wall) at zero temperature are described by a universal kernel, different from the universal edge kernel valid for smooth potentials.
• We compute this $d$ dimensional hard edge kernel exactly for a spherical domain and argue, using a generalized method of images, that it holds close to any sufficiently smooth boundary.
• As an application we compute the quantum statistics of the position of the fermion closest to the wall.