Ground State Energy of Dilute Bose Gas in Small Negative Potential Case

It is well known that the ground state energy of a three dimensional dilute Bose gas in the thermodynamic limit is $E=4\pi a \rho N$ when the particles interact via a non-negative, finite range, symmetric, two-body potential. Here, $N$ is the number of particles, $\rho$ is the density of the gas, and $a$ is the scattering length of the potential. In this paper, we prove the same result without the non-negativity condition on the potential, provided the negative part is small.