Interface asymptotics of Wigner—Weyl distributions for the Harmonic Oscillator

Boris Hanin*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We prove several types of scaling results for Wigner distributions of spectral projections of the isotropic Harmonic oscillator on ℝd. In prior work, we studied Wigner distributions Wh¯,EN(h¯)(x,ξ) of individual eigenspace projections. In this continuation, we study Weyl sums of such Wigner distributions as the eigenvalue EN(h¯) ranges over spectral intervals [E− δ(h¯) , E+ δ(h¯)] of various widths δ(h¯) and as (x, ξ) ∈ T*ℝd ranges over tubes of various widths around the classical energy surface Σ.E ⊂ T*ℝd. The main results pertain to interface Airy scaling asymptotics around ΣE, which divides phase space into an allowed and a forbidden region. The first result pertains to δ(h¯) = h¯ widths and generalizes our earlier results on Wigner distributions of individual eigenspace projections. Our second result pertains to δ(h¯) = h¯ 2 / 3 spectral widths and Airy asymptotics of the Wigner distributions in h¯ 2 / 3-tubes around ΣE. Our third result pertains to bulk spectral intervals of fixed width and the behavior of the Wigner distributions inside the energy surface, outside the energy surface and in a thin neighborhood of the energy surface.

Original languageEnglish (US)
JournalJournal d'Analyse Mathematique
DOIs
StateAccepted/In press - 2022

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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