Spectral properties of random Schrödinger operators are encoded in the average of products of Greens functions. For probability distributions with enough finite moments, the supersymmetric approach offers a useful dual representation. Here we use supersymmetric polar coordinates to derive a dual representation that holds for general distributions. We apply this result to study the density of states of the linearly correlated Lloyd model. In the case of non-negative correlation, we recover the well-known exact formula. In the case of linear small negative interaction localized around one point, we show that the density of states is well approximated by the exact formula. Our results hold on the lattice ℤ d $\mathbb {Z}^{d}$ uniformly in the volume.

中文翻译：

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超对称极坐标在劳埃德模型中的应用

随机薛定谔算子的光谱特性被编码为格林函数乘积的平均值。对于具有足够有限矩的概率分布，超对称方法提供了有用的对偶表示。在这里，我们使用超对称极坐标来导出适用于一般分布的对偶表示。我们应用这个结果来研究线性相关劳埃德模型的状态密度。在非负相关的情况下，我们恢复众所周知的精确公式。在局部于一个点的线性小负相互作用的情况下，我们表明状态密度很好地由精确公式近似。我们的结果在体积中均匀地保持在格子 ℤ d $\mathbb {Z}^{d}$ 上。