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

Abstract

In previous work we proved that the density of zeros of random eigenfunctions of the isotropic Harmonic Oscillator have different orders in the Planck constant h in the allowed and forbidden regions: In the allowed region the density is of order h^{-1} while it is h^{-1/2} in the forbidden region. In this article we study the density in the transition region around the caustic between the allowed and forbidden regions. Our main result is that the density is of order h^{-2/3} in an h^{2/3}-tube around the caustic. This tube radius is the 'critical radius'. For tubes of larger radius h^{alpha} with 0 < alpha < 2 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 h^{-2/3}. The asymptotics involve a fine study of the Airy scaling asymptotics of the eigenspace projections kernel near the caustic.