On the Inverse to the Harmonic Oscillator

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.