We study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.

中文翻译：

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超对称簇扩展及其在随机薛定ding算子中的应用

我们通过超对称形式主义研究离散随机Schrödinger算子。我们开发了在强和弱无序情况下都可以收敛的簇扩展。我们证明了无序平均格林函数的指数衰减以及弱扰动和在未扰动谱附近或强扰动和任何能量下的局部态密度的光滑度。作为一种应用，我们为状态的局部密度建立了Lifshitz尾型估计，从而确定了弱无序状态。