  Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$
perturbed by an isotropic pseudodifferential operator of order $1.$ We consider
the Schr\"odinger 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.
