D. Damanik and G. Teschl Bound States of Discrete Schr dinger Operators with Super-Critical Inverse Square Potentials (14K, LaTeX) ABSTRACT. We consider discrete one-dimensional Schr dinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of the number of eigenvalues below a given energy $E$ as this energy tends to the bottom of the essential spectrum.