Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on $\mathbb{Z}^2$

We study eigenvalues of discrete Schrödinger operators $H=Δ+V$ on $\mathbb{Z}^2$, where $Δ$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp