Schrödinger trace invariants for homogeneous perturbations of the harmonic oscillator

Moritz Doll, Steve Zelditch

Research output: Contribution to journalArticlepeer-review

Abstract

Let H = H0 + P denote the harmonic oscillator on Rd 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 ∈ 2π Z and the coefficients involve the dynamics of the Hamilton flow of the symbol σ(P) on the space CPd -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.

Original languageEnglish (US)
Pages (from-to)1303-1332
Number of pages30
JournalJournal of Spectral Theory
Volume10
Issue number4
DOIs
StatePublished - 2021

Keywords

  • Harmonic oscillator
  • Isotropic calculus
  • Threshold sub-principal perturbation
  • Trace formula

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Geometry and Topology

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