Title: Eigenstates and spectral projection for quantized baker's map Show Chinese title

Title: 本征态和量子化贝克映射的谱投影

Abstract: We extend the approach from [arXiv:2110.15301] to prove windowed spectral projection estimates and a generalized Weyl law for the (Weyl) quantized baker's map on the torus. The spectral window is allowed to shrink in the semiclassical (large dimension) limit. As a consequence, we obtain a strengthening of the quantum ergodic theorem from [arXiv:math-ph/0412058] to hold in shrinking spectral windows, a Weyl law on uniform spreading of eigenvalues, and statistics of random quasimodes. Using similar techniques, we also investigate random eigenbases of a different (non-Weyl) quantization, the Walsh-quantized baker's map, which has high degeneracies in its spectrum. For such random eigenbases, we prove that Gaussian eigenstate statistics and QUE hold with high probability in the semiclassical limit. Show Chinese abstract

Abstract: 我们将[arXiv:2110.15301]中的方法扩展，以证明在环面上的(Weyl)量子化Baker映射的窗口谱投影估计和广义Weyl定律。允许谱窗口在半经典(大维数)极限下缩小。作为结果，我们得到[arXiv:math-ph/0412058]中量子遍历定理的加强，在缩小的谱窗口中成立，一个关于本征值均匀扩散的Weyl定律，以及随机准模的统计特性。使用类似的技术，我们还研究了另一种(非Weyl)量子化方式的随机本征基，即Walsh量子化Baker映射，其谱具有高简并性。对于这样的随机本征基，我们在半经典极限下证明了高概率的高斯本征态统计和QUE成立。