Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator

Abstract

The expected hypersurface measure |$\mathcal {H}^{d-1}(Z_{E,h} \cap B(r, x))$| of nodal sets of random eigenfunctions of eigenvalue E of the semiclassical isotropic harmonic oscillator in balls |$B(r, x) \subset \mathbb {R}^d$| is determined as |$h{\rightarrow } 0.$| In the allowed region the volumes are of order h−1, while in the forbidden region they are of order h−1/2.