Seminar

We consider the quantum harmonic oscillator $H_0=(1/2)(-\Delta+|x|^2)$. The underlying classical flow is periodic with period $2\pi$. By an explicit calculation one can see that the Schrödinger propagator of $H_0$ is the identity (modulo a sign) at $2\pi\mathbb{Z}$ and locally smoothing otherwise. This periodicity is related to a sharp remainder estimate for the

counting function of the eigenvalues of $H_0$. If we perturb the operator by a pseudodifferential operator of lower order, then we break the symmetry and could hope for an improved remainder estimate. We will present results on recurrence of singularities for these operators as well as an improved remainder estimate. This is based on joint work with Oran Gannot, Jared Wunsch, and Steve Zelditch.