Optimal bounds for embedded eigenvalues of one-dimensional discrete Schrödinger operators with decaying potentials

In this paper, we consider one-dimensional discrete Schrödinger operators \begin{align} Hu(n)=(Δ+V)u(n)\nonumber \end{align} on ${\mathrm{\ell }}^2(\mathbb{N})$ with a self-adjoint boundary condition at $n=0$, where $\mathrm{\Delta }$ denotes the discrete Laplacian and $V(n)$ is a real-valued perturbation satisfying $$V(n)=\frac{O(1)}{1+n}\mathrm{.}$$ We determine the sharp transition for the asymptotic coefficient of $V$ governing the existence and nonexistence of embedded eigenvalues.