标题： 调和振子本征函数在焦散线附近的标度及其节点集 显示英文标题

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

摘要： 我们研究各向同性谐振子$- \hbar ^2 \Delta + |x|^2$的特征值$E = \hbar(N + \frac{d}{2})$的特征空间投影核$\Pi_{\hbar, E}(x,y)$在半经典极限$\hbar \to 0$中的标度渐近行为。 主要结果是给出了当$\hbar \to 0.$时，在焦散线$\mathcal C_E$的$\hbar^{2/3}$邻域内$\Pi_{\hbar, E}(x,y)$对$x,y$的缩放渐近的显式公式。缩放渐近被应用于高斯随机本征函数在焦散线附近的节点集分布，当$\hbar \to 0$时。 在之前的工作中，我们证明了高斯随机本征函数在$\hat{H}_{\hbar}$的零点密度在允许区域和禁止区域中具有不同的阶数，与普朗克常数$\hbar$相关：在允许区域中，密度为$\hbar^{-1}$阶，而在禁止区域中为$\hbar^{-1/2}$阶。 我们关于节点集的主要结果是，在焦散线周围的$\hbar^{\frac{2}{3}}$管中的零点密度为$\hbar^{-\frac{2}{3}}$阶。 该管的半径是临界半径。 对于内半径和外半径较大的环形区域$\hbar^{\alpha}$与$0< \alpha < \frac{2}{3}$，我们得到的密度结果在临界半径结果和我们在允许区域和禁止区域中的先前结果之间进行插值。 我们还证明了节点集与焦散线的交集的 Hausdorff$(d-2)$维测度为$\hbar^{- \frac{2}{3}}$阶。 显示英文摘要

摘要： We study the scaling asymptotics of the eigenspace projection kernels $\Pi_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 \Delta + |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $\Pi_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of $\hat{H}_{\hbar}$ have different orders in the Planck constant $\hbar$ in the allowed and forbidden regions: In the allowed region the density is of order $\hbar^{-1}$ while it is $\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that the density of zeros is of order $\hbar^{-\frac{2}{3}}$ in an $\hbar^{\frac{2}{3}}$-tube around the caustic. This tube radius is the `critical radius'. For annuli of larger inner and outer radii $\hbar^{\alpha}$ with $0< \alpha < \frac{2}{3}$ 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 $\hbar^{- \frac{2}{3}}$.