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arXiv · 1310.4532

Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator

Boris Hanin·Steve Zelditch·Peng Zhou
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Abstract

We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the semi-classical limit $h \to 0.$ In the allowed region the density is of order $h^{-1},$ while in the forbidden region the density is of order $h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this difference in "frequency" between the allowed and forbidden nodal sets.

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Boris Hanin, Steve Zelditch, Peng Zhou. 2013-10-16. Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator. https://arxiv.org/abs/1310.4532

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