Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Abstract

Let H = H 0 + P denote the harmonic oscillator on R d perturbed by an isotropic pseudodifferential operator P of order 1 and let U (t) = exp(−itH). We prove a Gutzwiller-Duistermaat-Guillemin type trace formula for Tr U (t). The singularities occur at times t ∈ 2πZ and the coefficients involve the dynamics of the Hamilton flow of the symbol σ(P ) on the space 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 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.

Recurrence of singularities for second order isotropic pseudodifferential operators

“…It is well-known (cf. Grigis-Sjöstrand [6,Chapter 11]) that the propagator is smoothing for ∉ ℤ and This was already proved in [4] using an explicit parametrix of the reduced propagator.…”

“…It is well‐known that implies that (cf. ). We refine this result slightly: Lemma Let and let be an open cone.…”

“…It is well-known that WF iso ( ) ∩ {0} × ℝ = ∅ implies that ∈  ∞ (cf. [4,8]). We refine this result slightly: 3…”

“…Proof. This follows from Lemma 3.1 in [4]. □ ) be real-valued and assume that there is a bounded set ⊂ SG 1,1 such that for all…”

“…We follow the notation of and denote the integral over the flow of by for any : The wavefront set of the reduced propagator can be completely determined: Proposition Let and . Then …”