发表机构
Université Paris-Saclay（巴黎-萨克雷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究紧流形上半经典Schrödinger算子随机扰动在混沌能量级附近的谱统计，证明一般扰动下特征值平滑计数函数的谱涨落在介观尺度上遵循随机矩阵理论预测的普适行为。
AI 中文摘要
我们研究了紧流形上半经典Schrödinger算子随机扰动的谱统计，在对应于混沌经典动力学的能量级附近。典型例子是算子$h^2\Delta+V(x)$受到光滑势$h^\alpha V_\omega$的扰动，其中$\alpha\in (0,1)$，$V_\omega$是距离$h^\beta$上解相关的随机势，$0<\beta<2\alpha$。我们证明对于一般扰动，特征值的平滑计数函数的谱涨落在某个介观尺度上遵循普适行为，这与随机矩阵理论的预测一致。
英文摘要
We investigate the spectral statistics of random perturbations of semiclassical Schrödinger operators on compact manifolds, near energy levels corresponding to chaotic classical dynamics. The prototypical example is that of an operator $h^2Δ+V(x)$ that is perturbed by a smooth potential $h^αV_ω$ with $α\in (0,1)$, and $V_ω$ is a random potential that decorrelates on distances $h^β$, with $0<β<2α$. We show that for a generic perturbation, the spectral fluctuations of the smoothed counting function of eigenvalues obey a universal behavior at a certain mesoscopic scale, which is coherent with the predictions of random matrix theory.
Comments46 pages, comments welcome
