标题： 调和振子本征函数在聚焦区附近的尺度化及其节点集 显示英文标题

标题： Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

摘要： 在之前的工作中，我们证明了各向同性谐振子随机本征函数零点的密度在允许区域和禁止区域中关于普朗克常数 h 的不同阶数：在允许区域中，密度为 h^{-1}的阶数，而在禁止区域中为 h^{-1/2}的阶数。 在本文中，我们研究了允许区域和禁止区域之间的奇点附近过渡区域中的密度。 我们的主要结果是，在奇点周围 h^{2/3}的管状区域内，密度为 h^{-2/3}的阶数。 这个管状区域的半径是“临界半径”。 对于半径更大的 h^{阿尔法}管状区域（其中 0 < alpha < 2），我们得到了在这一临界半径结果与我们在允许区域和禁止区域之前的结论之间插值的密度结果。 我们还证明了节点集与奇点的交集的 Hausdorff (d-2) 维测度为 h^{-2/3}的阶数。 渐近行为涉及对奇点附近本征空间投影核的 Airy 缩放渐近行为的细致研究。 显示英文摘要

摘要： In previous work we proved that the density of zeros of random eigenfunctions of the isotropic Harmonic Oscillator have different orders in the Planck constant h in the allowed and forbidden regions: In the allowed region the density is of order h^{-1} while it is h^{-1/2} in the forbidden region. In this article we study the density in the transition region around the caustic between the allowed and forbidden regions. Our main result is that the density is of order h^{-2/3} in an h^{2/3}-tube around the caustic. This tube radius is the 'critical radius'. For tubes of larger radius h^{alpha} with 0 < alpha < 2 we obtain density results which interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff (d-2)-dimensional measure of the intersection of the nodal set with the caustic is of order h^{-2/3}. The asymptotics involve a fine study of the Airy scaling asymptotics of the eigenspace projections kernel near the caustic.