A note on fractional powers of the Hermite operator

TL;DR

This paper provides a concise proof regarding the Weyl symbol of the inverse harmonic oscillator and extends these results to fractional powers, enhancing understanding of the operator's properties.

Contribution

It offers a simplified proof and extends existing results to fractional powers of the Hermite operator, advancing theoretical knowledge in this area.

Findings

Simplified proof of the Weyl symbol of the inverse harmonic oscillator

Extension of results to fractional powers of the Hermite operator

Enhanced theoretical understanding of operator properties

Abstract

We give a very short proof of a result proved by Cappiello-Rodino-Toft on the Weyl symbol of the inverse of the Harmonic oscillator. We also extend their results to fractional powers.