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

Abstract

We study eigenvalues of discrete Schrödinger operators $H=\Delta +V$ on ${\mathbb{Z}}^2$, where $\Delta$ 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 decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge $\lambda =\pm 4$, and the bulk, there is a further distinction between the regular energies $0<|\lambda |<4$ and the interior critical energy $\lambda =0$ stemming from the reducibility of the corresponding Fermi surface in the latter case. For every $0<|\lambda |<4$, we construct potentials $V$ satisfying $|V(n)|\le C|n{|}^{-1}$ for which $\lambda$ is an eigenvalue of $\Delta +V$, and prove absence of eigenvalues when $|V(n)|\le C|n{|}^{-1-\epsilon }$ for some $\epsilon >0$. At $\lambda =0$, the critical power changes and we construct potentials $V$ satisfying $|V(n)|\le C|n{|}^{-2}$ for which $0$ is an eigenvalue of $\Delta +V$, as well as prove absence when $|V(n)|\le C|n{|}^{-2-\epsilon }$ for any $\epsilon >0$. Finally, at each spectral edge, we show that, for every $K\ge 3$, an eigenvalue can be created by potentials supported on exactly $K$ sites, whereas a potential supported on at most two sites cannot create an edge eigenvalue. The proofs combine Green-function expansions and moment cancellation, Hilbert-space-valued iterations, discrete Carleman estimates, and a uniform Green-kernel estimate.