Which pseudodifferential operators with radial symbols are non-negative?

Abstract

Operators the Weyl symbols of which are of the special kind \(f(x,\xi )=F(|x|^2+|\xi |^2)\) are in the spectral-theoretic sense functions \(\Psi \left(L^{(n)}\right)\) of the harmonic oscillator \(L^{(n)}\). We give the exact conditions on the real function F which ensure that the function \(\Psi \) (defined on the set \(\{\frac{n}{2}+k,\,k=0,1,\dots \}\)) will be non-negative, non-decreasing or convex. In the one-dimensional case, the first point can also be treated within a one-parameter relativistic deformation of the whole situation, involving in place of Hermite functions the no longer elementary (modified) Mathieu functions.