A long time Gutzwiller trace formula for perturbations of Schrödinger operators near Anosov energy levels

Let $P_h$ be a semiclassical pseudodifferential operator on a closed Riemannian manifold $(M,g)$, with principal symbol $p$. We consider perturbations $P_h+h^αQ$, with $0<α\le 1$, where $Q$ is a semiclassical pseudodifferential operator whose symbol oscillates at scale $h^β$, with $2β<α$. Assuming t