We consider the discrete Schrödinger operator \(H=-\Delta +V\) on a cube \(M\subset {{\mathbb {Z}}}^d\), with periodic or Dirichlet (simple) boundary conditions. We use a hidden landscape function u, defined as the solution of an inhomogeneous boundary problem with uniform right-hand side for H, to predict the location of the localized eigenfunctions of H. Explicit bounds on the exponential decay of Agmon type for low-energy modes are obtained. This extends the recent work of Agmon type of localization in Arnold et al. (Commun Partial Differ Equ 44:1186–1216, 2019) for \({\mathbb {R}}^d\) to a tight-binding Hamiltonian on \({{\mathbb {Z}}}^d\) lattice. Contrary to the continuous case, high-energy modes are as localized as the low-energy ones in discrete lattices. We show that exponential decay estimates of Agmon type also appear near the top of the spectrum, where the location of the localized eigenfunctions is predicted by a different landscape function. Our results are deterministic and are independent of the size of the cube. We also provide numerical experiments to confirm the conditional results effectively, for some random potentials.

我们考虑具有周期或Dirichlet（简单）边界条件的立方体\（M \ subset {{\ mathbb {Z}}} ^ d \）上的离散Schrödinger运算符\（H =-\ Delta + V \）。我们使用隐式景观函数u定义为H的局部本征函数的位置，该隐函数定义为H的右侧均匀的非均匀边界问题的解。获得了低能模式下Agmon型指数衰减的显式界。这扩展了Arnold等人最近关于Agmon型定位的工作。（公共偏差方程44：1186–1216，2019）为\（{\ mathbb {R}} ^ d \）到\（{{\ mathbb {Z}}} ^ d \格子。与连续情况相反，高能模式在离散晶格中的位置与低能模式一样。我们表明，Agmon类型的指数衰减估计值也出现在频谱的顶部附近，其中局部本征函数的位置由不同的景观函数预测。我们的结果是确定性的，并且与多维数据集的大小无关。我们还提供了一些数值实验，以针对某些随机电位有效地确认条件结果。