发表机构
UC Berkeley; Texas A&M University（加州大学伯克利分校; 德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了 $\mathbb{Z}^2$ 上离散薛定谔算子在不同谱区域（内部、临界点、边缘）特征值存在的尖锐衰减阈值，并给出构造与不存在性证明。
AI 中文摘要
我们研究 $\mathbb{Z}^2$ 上离散薛定谔算子 $H=\Delta+V$ 的特征值，其中 $\Delta$ 是非中心拉普拉斯算子，即 $\mathbb{Z}^2$ 的未归一化邻接算子，且 $V$ 在无穷远处衰减。根据外尔定理，$H$ 的本质谱为 $[-4,4]$。我们确定了三个不同谱区域内特征值存在的尖锐衰减阈值。虽然在谱边缘 $\lambda=\pm4$ 和谱内部自然预期会有不同行为，但正则能量 $0<|\lambda|<4$ 与内部临界能量 $\lambda=0$ 之间还存在进一步的区别，后者源于相应费米面的可约性。对于每个 $0<|\lambda|<4$，我们构造满足 $|V(n)|\leq C|n|^{-1}$ 的势 $V$，使得 $\lambda$ 是 $\Delta+V$ 的特征值，并证明当 $|V(n)|\leq C|n|^{-1-\varepsilon}$（对某个 $\varepsilon>0$）时不存在特征值。在 $\lambda=0$ 处，临界幂次发生变化，我们构造满足 $|V(n)|\leq C|n|^{-2}$ 的势 $V$，使得 $0$ 是 $\Delta+V$ 的特征值，并证明当 $|V(n)|\leq C|n|^{-2-\varepsilon}$（对任意 $\varepsilon>0$）时不存在特征值。最后，在每个谱边缘，我们证明对于每个 $K\geq3$，可以由恰好支撑在 $K$ 个位点上的势产生特征值，而支撑在至多两个位点上的势不能产生边缘特征值。证明结合了格林函数展开和矩消去、希尔伯特空间值迭代、离散卡尔曼估计以及一致格林核估计。
英文摘要
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 decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge $λ=\pm4$, and the bulk, there is a further distinction between the regular energies $0<|λ|<4$ and the interior critical energy $λ=0$ stemming from the reducibility of the corresponding Fermi surface in the latter case. For every $0<|λ|<4$, we construct potentials $V$ satisfying $|V(n)|\leq C|n|^{-1}$ for which $λ$ is an eigenvalue of $Δ+V$, and prove absence of eigenvalues when $|V(n)|\leq C|n|^{-1-\varepsilon}$ for some $\varepsilon>0$. At $λ=0$, the critical power changes and we construct potentials $V$ satisfying $|V(n)|\leq C|n| ^{-2}$ for which $0$ is an eigenvalue of $Δ+V$, as well as prove absence when $|V(n)|\leq C|n|^{-2-\varepsilon}$ for any $\varepsilon>0$. Finally, at each spectral edge, we show that, for every $K\geq3$, 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.
CommentsMinor improvements to exposition; no changes to mathematical content
