2019/01/18 by Boris Hanin, Steve Zelditch, Hanin, Boris +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1901.06438
openalex publication_date 2019/01/18 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28
Eigenspaces of the quantum isotropic Harmonic Oscillator \H\ℏ : =\n- \(\ℏ2)/(2) \Δ + \(||x||2)/(2) on \ℝd have\nextremally high multiplicites and the eigenspace projections \Π\ℏ,\nEN(\ℏ) have special asymptotic properties. This article gives a detailed\nstudy of their Wigner distributions W\ℏ, EN(\ℏ)(x, \ξ).\nHeuristically, if EN(\ℏ) = E, W\ℏ, EN(\ℏ)(x, \ξ) is the\n`quantization' of the energy surface \ΣE, and should be like the\ndelta-function \δ\ΣE on \ΣE; rigorously, W\ℏ,\nEN(\ℏ)(x, \ξ) tends in a weak* sense to \δ\ΣE. But its\npointwise asymptotics and scaling asymptotics have more structure. The main\nresults give Bessel asymptotics of W\ℏ, EN(\ℏ)(x, \ξ) in the\ninterior H(x, \ξ) < E of \ΣE; interface Airy scaling asymptotics in\ntubes of radius \ℏ2/3 around \ΣE, with (x, \ξ) either in the\ninterior or exterior of the energy ball; and exponential decay rates in the\nexterior of the energy surface.\n