## Abstract

We study the scaling asymptotics of the eigenspace projection kernels Π _{ħ} _{,} _{E}(x, y) of the isotropic Harmonic Oscillator H^ _{ħ}= - ħ^{2}Δ + | 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.

Original language | English (US) |
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Pages (from-to) | 1147-1183 |

Number of pages | 37 |

Journal | Communications In Mathematical Physics |

Volume | 350 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2017 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics