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

We study the scaling asymptotics of the eigenspace projection kernels Π _{ħ , E}(x, y) of the isotropic Harmonic Oscillator H^ _ħ= - ħ²Δ + | x| ² of eigenvalue E=ħ(N+d2) in the semi-classical limit ħ→ 0 . The principal result is an explicit formula for the scaling asymptotics of Π _{ħ , E}(x, y) for x, y in a ħ^{2 / 3} neighborhood of the caustic C_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 H^ _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^{- 1} while it is ħ^{- 1 / 2} in the forbidden region. Our main result on nodal sets is that the density of zeros is of order ħ-23 in an ħ23 -tube around the caustic. This tube radius is the ‘critical radius’. For annuli of larger inner and outer radii ħ^α with 0<α<23 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-23.