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

TL;DR

This paper derives explicit scaling asymptotics for harmonic oscillator eigenfunctions near the caustic and applies these results to analyze the distribution and density of nodal sets of Gaussian eigenfunctions in the semi-classical limit.

Contribution

It provides a new explicit formula for the eigenfunction kernel asymptotics near the caustic and characterizes the nodal set density and measure in this critical region.

Findings

Nodal set density is of order $sh^{-2/3}$ near the caustic

Hausdorff measure of nodal intersections with the caustic is of order $sh^{-2/3}$

Scaling asymptotics are established for eigenfunction kernels in a $sh^{2/3}$ neighborhood of the caustic.

Abstract

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 $\hbar^{- \frac{2}{3}}$.