论文标题
通过景观和双层景观功能，特征性的征算的指数衰减
The exponential decay of eigenfunctions for tight binding Hamiltonians via landscape and dual landscape functions
论文作者
论文摘要
我们考虑了Cube $ M \ subset \ Mathbb {z}^d $上的离散schrödinger运算符$ h =-Δ+v $，带有定期或dirichlet（简单）边界条件。我们使用隐藏的景观函数$ u $，定义为$ h $的不均匀边界问题的解决方案，以预测$ h $的局部特征函数的位置。获得低能模式的Agmon类型的指数衰减的明确界限。这将$ \ mathbb {r}^d $在[5]中的Agmon类型本地化的最新工作扩展到了$ \ Mathbb {z}^d $ lattice上的紧密结合的汉密尔顿。与连续的情况相反，高能模式与离散晶格中的低能能量一样局部。我们表明，Agmon类型的指数衰减估计也出现在光谱顶部附近，在该频谱的顶部，在该景观中，通过不同的景观功能预测了局部本征函数的位置。我们的结果是确定性的，并且与立方体的大小无关。我们还提供数值实验，以有效地确认一些随机电位的条件结果。
We consider the discrete Schrödinger operator $H=-Δ+V$ 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 [5] 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.
