Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator

Boris Hanin, Steve Zelditch

Research output: Contribution to journalArticlepeer-review


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
Issue number11
StatePublished - Nov 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


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


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