# Interface Asymptotics of Wigner-Weyl Distributions for the Harmonic Oscillator

## TL;DR

This paper investigates the asymptotic behavior of Wigner distributions for the harmonic oscillator's spectral projections, revealing detailed interface phenomena around the classical energy surface with various spectral interval widths.

## Contribution

It extends previous work by analyzing Weyl sums of Wigner distributions over spectral intervals and derives new Airy scaling asymptotics near the energy surface for different spectral widths.

## Key findings

- Derived Airy asymptotics for spectral widths of b4(\u210f) = b4() and b4() = b4()^{2/3} around the energy surface.
- Established behavior of Wigner distributions inside, outside, and near the energy surface for fixed spectral intervals.
- Generalized earlier results on individual eigenspace projections to Weyl sums over spectral intervals.

## Abstract

We prove several types of scaling results for Wigner distributions of spectral projections of the isotropic Harmonic oscillator on $\mathbb R^d$. In prior work, we studied Wigner distributions $W_{\hbar, E_N(\hbar)}(x, \xi)$ of individual eigenspace projections. In this continuation, we study Weyl sums of such Wigner distributions as the eigenvalue $E_N(\hbar)$ ranges over spectral intervals $[E - \delta(\hbar), E + \delta(\hbar)]$ of various widths $\delta(\hbar)$ and as $(x, \xi) \in T^*\mathbb R^d$ ranges over tubes of various widths around the classical energy surface $\Sigma_E \subset T^*\mathbb R^d$. The main results pertain to interface Airy scaling asymptotics around $\Sigma_E$, which divides phase space into an allowed and a forbidden region. The first result pertains to $\delta(\hbar) = \hbar$ widths and generalizes our earlier results on Wigner distributions of individual eigenspace projections. Our second result pertains to $\delta(\hbar) = \hbar^{2/3}$ spectral widths and Airy asymptotics of the Wigner distributions in $\hbar^{2/3}$-tubes around $\Sigma_E$. 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.