Interface asymptotics of Wigner—Weyl distributions for the Harmonic Oscillator

• Published in: Journal d'analyse mathématique (Jerusalem) 2022-10, Vol.147 (1), p.69-98
• Main Authors: Hanin, Boris, Zelditch, Steve
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Analysis; Asymptotic properties; Dynamical Systems and Ergodic Theory; Eigenvalues; Functional Analysis; Harmonic oscillators; Intervals; Mathematics; Mathematics and Statistics; Partial Differential Equations; Tubes; Wigner distribution
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-022-0209-4

We prove several types of scaling results for Wigner distributions of spectral projections of the isotropic Harmonic oscillator on ℝ d . In prior work, we studied Wigner distributions W h ¯ , E N ( h ¯ ) ( x , ξ ) of individual eigenspace projections. In this continuation, we study Weyl sums of such Wigner distributions as the eigenvalue E N ( h ¯ ) ranges over spectral intervals [ E − δ ( h ¯ ) , E + δ ( h ¯ ) ] of various widths δ ( h ¯ ) and as ( x, ξ ) ∈ T *ℝ d ranges over tubes of various widths around the classical energy surface Σ. E ⊂ T *ℝ d . The main results pertain to interface Airy scaling asymptotics around Σ E , 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 Σ E . 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.