Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Let H denote the harmonic oscillator Hamiltonian on R d , perturbed by an isotropic pseudodifferential operator of order 1. We consider the Schrödinger propagator U ( t ) = e - i t H , and find that while sing - supp Tr U ( t ) ⊂ 2 π Z as in the unperturbed case, there exists a large class of pertur...

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Bibliographic Details
Published inCommunications in mathematical physics Vol. 362; no. 1; pp. 269 - 294
Main Authors Doll, Moritz, Gannot, Oran, Wunsch, Jared
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2018
Springer Nature B.V
Subjects
Classical and Quantum Gravitation
Complex Systems
Harmonic oscillators
Mathematical and Computational Physics
Mathematical Physics
Oscillators
Physics
Physics and Astronomy
Quantum Physics
Quantum theory
Relativity Theory
Singularity (mathematics)
Theoretical
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Summary:Let H denote the harmonic oscillator Hamiltonian on R d , perturbed by an isotropic pseudodifferential operator of order 1. We consider the Schrödinger propagator U ( t ) = e - i t H , and find that while sing - supp Tr U ( t ) ⊂ 2 π Z as in the unperturbed case, there exists a large class of perturbations in dimensions d ≥ 2 for which the singularities of Tr U ( t ) at nonzero multiples of 2 π are weaker than the singularity at t  = 0. The remainder term in the Weyl law is of order o ( λ d - 1 ) , improving in these cases the o ( λ d - 1 ) remainder previously established by Helffer–Robert.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-018-3100-5