Superscars for arithmetic point scatters II
Superscars for arithmetic point scatters II
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DOI：
10.1017/fms.2023.33
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
P. Kurlberg;S. Lester;Lior Rosenzweig
中科院分区：
文献类型：
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作者：
P. Kurlberg;S. Lester;Lior Rosenzweig
摘要我们考虑标准平坦环面上点散射体本征函数的量子极限（半经典测度）测度的动量推进。给定任何概率测度，通过将δ质量以相等的权重放置在圆上的${\mathbb Z}^2$ -格点上并投影到单位圆，我们证明了本征函数的某些连续性的质量，在动量空间中，完全局部化在该测度上，并且在位置上完全离域（即，拉格朗日状态（Lagrangian states）我们还表明，质量，在动量，可以完全本地化更奇异的措施，例如，奇异连续的支持康托集。此外，我们可以给出量子极限的例子，这些量子极限是这些测度的某些凸组合，特别是表明量子极限的集合比仅由圆上格点的弱极限产生的量子极限更丰富。证明利用特征的半维筛和行为的乘法函数在短时间内，使扰动特征值的位置的精确控制。
Abstract We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus ${\mathbb T}^2 = {\mathbb R}^2/{\mathbb Z}^{2}$ . Given any probability measure arising by placing delta masses, with equal weights, on ${\mathbb Z}^2$ -lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.