## Abstract

Eigenspaces of the quantum isotropic Harmonic Oscillator (Formula presented.) on (Formula presented.) have extremally high multiplicites and the eigenspace projections (Formula presented.) have special asymptotic properties. This article gives a detailed study of their Wigner distributions (Formula presented.) Heuristically, if (Formula presented.) is the “quantization” of the energy surface Σ_{E}, and should be like the delta-function (Formula presented.) on Σ_{E}; rigorously, (Formula presented.) tends in a weak* sense to (Formula presented.) But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of (Formula presented.) in the interior (Formula presented.) of Σ_{E}; interface Airy scaling asymptotics in tubes of radius (Formula presented.) around Σ_{E}, with (Formula presented.) either in the interior or exterior of the energy ball; and exponential decay rates in the exterior of the energy surface.

Original language | English (US) |
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Pages (from-to) | 1589-1620 |

Number of pages | 32 |

Journal | Communications in Partial Differential Equations |

Volume | 45 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2020 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

## Keywords

- Harmonic oscillator
- Wigner function
- semiclassical analysis
- spectral theory