Refined Weyl law for homogeneous perturbations of the harmonic oscillator

Let $H$ denote the harmonic oscillator Hamiltonian on $mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrodinger propagator $U(t)=e^{-itH},$ and find that while $operatorname{singsupp} operatorname{Tr} U(t) subset 2 pi mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d geq 2$ for which the singularities of $operatorname{Tr} U(t)$ at nonzero multiples of $2 pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(lambda^{d-1})$, improving in these cases the $O(lambda^{d-1})$ remainder previously established by Helffer--Robert.