Schrödinger Trace Invariants for Homogeneous Perturbations of the Harmonic Oscillator

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 singular...

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Bibliographic Details
Published inarXiv.org
Main Authors Doll, Moritz, Zelditch, Steve
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.11.2018
Subjects
Harmonic oscillators
Mathematical analysis
Quantum theory
Singularities
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Summary: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.
ISSN:2331-8422