In the presence of a confining potential V, the eigenfunctions of a continuous Schrödinger operator −Δ + V decay exponentially with the rate governed by the part of V, which is above the corresponding eigenvalue; this can be quantified by a method of Agmon. Analogous localization properties can also be established for the eigenvectors of a discrete Schrödinger matrix. This note shows, perhaps surprisingly, that one can replace a discrete Schrödinger matrix by any real symmetric Z-matrix and still obtain eigenvector localization estimates. In the case of a real symmetric non-singular M-matrix A (which is a situation that arises in several contexts, including random matrix theory and statistical physics), the landscape function u = A⁻¹1 plays the role of an effective potential of localization. Starting from this potential, one can create an Agmon-type distance function governing the exponential decay of the eigenfunctions away from the “wells” of the potential, a typical eigenfunction being localized to a single such well.

中文翻译：

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anM矩阵的有效潜力

在存在限制电位V的情况下，连续Schrödinger算子-Δ+ V的本征函数以由V的部分控制的速率呈指数衰减，该比率高于相应的本征值。这可以通过Agmon的方法来量化。还可以为离散的薛定矩阵的特征向量建立类似的定位特性。也许令人惊讶的是，这一注释表明，可以用任何实对称Z矩阵代替离散的Schrödinger矩阵，并且仍然可以获得特征向量定位估计。在实对称非奇异M-矩阵A的情况下（这是在多种情况下出现的情况，包括随机矩阵理论和统计物理学），景观函数 u = A ^-1 1发挥了有效的局部化作用。从这一势能开始，可以创建一个Agmon型距离函数，以控制本征函数的指数衰减，使其远离势能的“井”，典型的本征函数局限于一个这样的井。