Refined Weyl law for homogeneous perturbations of the harmonic oscillator

Abstract

Let $H$ denote the harmonic oscillator Hamiltonian on ${\mathbb{R}}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrödinger propagator $U(t)=e^{-itH},$ and find that while $\mathrm{singsupp}\mathrm{Tr}U(t)\subset 2\pi \mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d\ge 2$ for which the singularities of $\mathrm{Tr}U(t)$ at nonzero multiples of $2\pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o({\lambda }^{d-1})$, improving in these cases the $O({\lambda }^{d-1})$ remainder previously established by Helffer--Robert.

28 pages; new section added on propagation of singularities

Cited by in corpus (3)

• Constants of Motion of the Harmonic Oscillator
• A Shubin pseudodifferential calculus on asymptotically conic manifolds
• Recurrence of singularities for second order isotropic pseudodifferential operators