Quantum Limits for Harmonic Oscillator

TL;DR

This paper investigates the behavior of high energy eigenfunctions of the harmonic oscillator in multiple dimensions, demonstrating that any invariant measure on the energy surface can be approximated as a weak limit of these eigenfunctions.

Contribution

It establishes a connection between invariant measures on the energy surface and high energy eigenfunctions, showing they can approximate each other in the weak limit.

Findings

Any invariant measure on the energy surface can be realized as a weak limit of eigenfunctions.

The results extend understanding of eigenfunction concentration in quantum harmonic oscillators.

Provides a framework for approximating invariant measures using eigenfunctions.

Abstract

In this note we consider high energy eigenfunctions of the harmonic oscillator in $\mathbb{R}^d$ and prove that any invariant measure on the energy surface can be written as a weak limit of eigenfunctions.