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

Abstract

We study the scaling asymptotics of the eigenspace projection kernels ${\Pi_{\hbar, E}(x,y)}$ of the isotropic Harmonic Oscillator ${\hat{H}_{\hbar} = - \hbar^2 \Delta +|x|^2}$ 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 ${\hbar \rightarrow 0.}$ 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 that 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 ${\hbar^{- \frac{2}{3}}}$ .