Schr\"odinger Trace Invariants for Homogeneous Perturbations of the Harmonic Oscillator

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 $\mathrm{Tr}U(t)$. 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 ${\mathbb{C}\mathbb{P}}^{d-1}$ 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 $\mathrm{Tr}U(t)$. 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.