Some properties of threshold eigenstates and resonant states of discrete Schr\"odinger operators

In this note, we study some properties of threshold resonant states or threshold eigenfunctions for discrete Schr"odinger operators. We mainly prove two theorems. First, we prove that resonant states at the elliptic threshold have the same asymptotic expansion as the continuous Schr"odinger operator. Second, we prove absence of resonant states at hyperbolic thresholds