Gradient corrections to the local density approximation for trapped superfluid Fermi gases

Two species superfluid Fermi gas is investigated on the BCS side up to the Feshbach resonance. Using the Greens's function technique gradient corrections are calculated to the generalized Thomas-Fermi theory including Cooper pairing. Their relative magnitude is found to be measured by the small parameter $(d/R_{TF})^4$, where $d$ is the oscillator length of the trap potential and $R_{TF}$ is the radial extension of the density $n$ in the Thomas-Fermi approximation. In particular at the Feshbach resonance the universal %constant $A_{TF}$ has the %correction in the center $A=A_{TF}+A_2(d/R_{TF})^4+\...$ corrections to the local density approximation are calculated and a universal prefactor $\kappa_W=7/27$ is derived for the von Weizs\"acker type correction $\kappa_W(\hbar^2/2m)(\nabla^2 n^{1/2}/n^{1/2})$.