Schrödinger trace invariants for homogeneous perturbations of the harmonic oscillator

Abstract

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 \operatorname{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{CP}^{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 \operatorname{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.

Locations

• Journal of Spectral Theory - View
• arXiv (Cornell University) - View - PDF