Recently,M. and T. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the density of states.

We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations of superspace, and was applied earlier in the context of band matrices by Constantinescu. Other versions of this supersymmetry have been a crucial ingredient in the study of the localization–delocalization transition by theoretical physicists.

中文翻译：

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一维随机带矩阵通过超对称转移算子的状态密度

最近，M。T. Shcherbina和T. Shcherbina证明了大带宽一维高斯带矩阵的状态密度的逐点半圆定律。他们证明的主要步骤是研究半经典状态下非自伴算子的光谱特性的新方法。该方法适用于由状态密度的超对称积分表示构造的传递算子。

我们为半圆定律的稍微升级版本提供了一个更简单的证明，该定律仅需要标准的半经典论点和一些特殊的基本计算即可。简化归因于超对称性的使用，超对称性的出现表现在传递算符与超空间变换族之间的交换中，并由Constantinescu在频带矩阵的上下文中得到了较早的应用。这种超对称性的其他版本已成为理论物理学家研究本地化至非本地化转变的重要组成部分。