Spectral gap for the Kac model with hard sphere collisions

We prove the analog of the Kac conjecture for hard sphere collisions, giving a computable, close estimate on the spectral gap that is independent of the number of particles. Previous work has focused on the case in which the collision rates are independent of the particle velocities, the case of so-called Maxwellian molecules. The new methods introduced here allow us to deal with collision rates that are not bounded from below. We also obtain information on the structure of the gap eigenfunction.