Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator

Boris Hanin; Steve Zelditch

doi:10.1080/03605302.2020.1784208

Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator

Boris Hanin^*, Steve Zelditch

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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)
Pages (from-to)	1589-1620
Number of pages	32
Journal	Communications in Partial Differential Equations
Volume	45
Issue number	11
DOIs	https://doi.org/10.1080/03605302.2020.1784208
State	Published - Nov 1 2020

Funding

BH is supported by NSF grants DMS?1855684 and CCF?1934904 and SZ is partially supported by NSF grant DMS?1810747.

Keywords

Harmonic oscillator
Wigner function
semiclassical analysis
spectral theory

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1080/03605302.2020.1784208

Cite this

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title = "Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator",

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.",

keywords = "Harmonic oscillator, Wigner function, semiclassical analysis, spectral theory",

author = "Boris Hanin and Steve Zelditch",

note = "Funding Information: BH is supported by NSF grants DMS?1855684 and CCF?1934904 and SZ is partially supported by NSF grant DMS?1810747. Publisher Copyright: {\textcopyright} 2020 Taylor & Francis Group, LLC.",

year = "2020",

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doi = "10.1080/03605302.2020.1784208",

language = "English (US)",

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T1 - Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator

AU - Hanin, Boris

AU - Zelditch, Steve

PY - 2020/11/1

Y1 - 2020/11/1

N2 - 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.

AB - 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.

KW - Harmonic oscillator

KW - Wigner function

KW - semiclassical analysis

KW - spectral theory

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JO - Communications in Partial Differential Equations

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Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator

Abstract

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Keywords

ASJC Scopus subject areas

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