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

Abstract

We study the scaling asymptotics of the eigenspace projection kernels Π[subscript ℏ,E](x,y) of the isotropic Harmonic Oscillator [^ over H][subscript ℏ]=−ℏ[superscript 2]Δ+|x|[superscript 2] of eigenvalue E=ℏ(N+d/2) in the semi-classical limit ℏ→0. The principal result is an explicit formula for the scaling asymptotics of Π[subscript ℏ,E](x,y) for x, y in a ℏ[superscript 2/3] neighborhood of the caustic C[subscript E] as ℏ→0. The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as ℏ→0. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of [^ over H][subscript ℏ] have different orders in the Planck constant ℏ in the allowed and forbidden regions: In the allowed region the density is of order ℏ[superscript −1] while it is ℏ[superscript −1/2] in the forbidden region. Our main result on nodal sets is that the density of zeros is of order ℏ[superscript −2/3] in an ℏ[superscript 2/3] -tube around the caustic. This tube radius is the ‘critical radius’. For annuli of larger inner and outer radii ℏ[superscript α] with 0<α<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 ℏ[superscript −2/3] .