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 $P_h+h^{\alpha }Q$, with $0<\alpha \le 1$, where $Q$ is a semiclassical pseudodifferential operator whose symbol oscillates at scale $h^{\beta }$, with $2\beta <\alpha$. Assuming that the classical dynamics of the unperturbed system on a regular energy surface $p^{-1}(E)$ is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order $\mathrm{\mid }\log h\mathrm{\mid }$. This formula relates the distribution of energy levels of the quantum system near the energy $E$ to the periodic orbits of its classical counterpart. The perturbation $h^{\alpha }Q$ affects only the phases appearing in the trace formula.