  Let $H = H_0 + P$ denote the harmonic oscillator on $\mathbb{R}^d$ perturbed
by an isotropic pseudodifferential operator $P$ of order $1$ and let $U(t) =
\operatorname{exp}(- it H)$. We prove a Gutzwiller-Duistermaat-Guillemin type
trace formula for $\operatorname{Tr} U(t)$. The singularities occur at times $t
\in 2 \pi \mathbb{Z}$ and the coefficients involve the dynamics of the Hamilton
flow of the symbol $\sigma(P)$ on the space $\mathbb{CP}^{d-1}$ of harmonic
oscillator orbits of energy $1$. This is a novel kind of sub-principal symbol
effect on the trace. We generalize the averaging technique of Weinstein and
Guillemin to this order of perturbation, and then present two completely
different calculations of $\operatorname{Tr} U(t)$. The first proof directly
constructs a parametrix of $U(t)$ in the isotropic calculus, following earlier
work of Doll-Gannot-Wunsch. The second proof conjugates the trace to the
Bargmann-Fock setting, the order $1$ of the perturbation coincides with the
`central limit scaling' studied by Zelditch-Zhou for Toeplitz operators.
