An exact semiclassical version of the classical KAM theorem about small perturbations of vector fields on the torus is given. Moreover, a renormalization theorem based on counterterms for some semiclassical systems that are close to being completely integrable is obtained. We apply these results to characterize the sets of semiclassical measures and quantum limits for sequences of $L^2$-eigenfunctions of these systems.

中文翻译：

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重整化 KAM 系统的谱稳定性和半经典测度

给出了关于环面上向量场的小扰动的经典 KAM 定理的精确半经典版本。此外，对于一些接近完全可积的半经典系统，得到了基于反项的重整化定理。我们应用这些结果来表征这些系统的 $L^2$-特征函数序列的半经典测度集和量子极限。