Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator

Boris Hanin, Steve Zelditch

Research output: Contribution to journalArticlepeer-review

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 ΣE, and should be like the delta-function (Formula presented.) on ΣE; 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 ΣE; interface Airy scaling asymptotics in tubes of radius (Formula presented.) around ΣE, 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 languageEnglish (US)
Pages (from-to)1589-1620
Number of pages32
JournalCommunications in Partial Differential Equations
Volume45
Issue number11
DOIs
StatePublished - Nov 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Keywords

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

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