On the inverse to the harmonic oscillator

TL;DR

This paper derives explicit formulas and bounds for the Weyl symbol of the inverse harmonic oscillator, revealing its properties in Gevrey and Gelfand-Shilov classes, especially in even dimensions.

Contribution

It provides explicit expressions and bounds for the inverse harmonic oscillator's Weyl symbol, utilizing symmetry and advanced analytical techniques.

Findings

Bounds of Gevrey and Gelfand-Shilov type for the symbol

Explicit formulas in even dimensions

Characterization using elementary functions

Abstract

Let $b_d$ be the Weyl symbol of the inverse to the harmonic oscillator on $\R^d$. We prove that $b_d$ and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for $b_d$. In the even-dimensional case we characterize $b_d$ in terms of elementary functions. In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions.