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The expected hypersurface measure $\cal {H}^{d-1}(Z_{E,h} \cap B(r, x))$ of nodal sets of random eigenfunctions of eigenvalue E of the semiclassical isotropic harmonic oscillator in balls $B(r, x) \subset {\BBR}^d$ 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.

Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator