2019/03/29 by Boris Hanin, Steve Zelditch, Hanin, Boris +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1903.12524
openalex publication_date 2019/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove several types of scaling results for Wigner distributions of\nspectral projections of the isotropic Harmonic oscillator on mathbb Rd. In\nprior work, we studied Wigner distributions W\ℏ, EN(\ℏ)(x, \ξ) of\nindividual eigenspace projections. In this continuation, we study Weyl sums of\nsuch Wigner distributions as the eigenvalue EN(\ℏ) ranges over spectral\nintervals [E - \δ(\ℏ), E + \δ(\ℏ)] of various widths\n\δ(\ℏ) and as (x, \ξ) \∈ T^* mathbb Rd ranges over tubes of\nvarious widths around the classical energy surface \ΣE \⊂ T^* mathbb\nRd. The main results pertain to interface Airy scaling asymptotics around\n\ΣE, which divides phase space into an allowed and a forbidden region.\nThe first result pertains to \δ(\ℏ) = \ℏ widths and generalizes our\nearlier results on Wigner distributions of individual eigenspace projections.\nOur second result pertains to \δ(\ℏ) = \ℏ2/3 spectral widths and\nAiry asymptotics of the Wigner distributions in \ℏ2/3-tubes around\n\ΣE. Our third result pertains to bulk spectral intervals of fixed width\nand the behavior of the Wigner distributions inside the energy surface, outside\nthe energy surface and in a thin neighborhood of the energy surface.\n