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The semiclassical limit from Hartree to Vlasov at positive density: strong uniform-in-time convergence and scattering

  • Marnie Smith
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Abstract

Strong semiclassical convergence from the Hartree equation to the Vlasov equation is established in three dimensions near Penrose-stable homogeneous steady states at positive density. For sufficiently regular integrable interaction kernels, an O(ℏ2) convergence rate in weighted Sobolev spaces is proved on finite time intervals. Combining this estimate with earlier uniform-in-ℏ phase-mixing and scattering bounds for the Hartree equation yields global existence and scattering for the corresponding Vlasov solution through the semiclassical limit. If the quantum and classical initial perturbations are aligned, the semiclassical convergence is uniform for all times, and the Wigner transforms of the quantum scattering profiles converge to the classical scattering profile at the explicit rate O((lnℏ−1)−1/3).

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Cite

BibTeXRIS
BibTeX
@misc{smith2026semiclassical,
  title = {{The semiclassical limit from Hartree to Vlasov at positive density: strong uniform-in-time convergence and scattering}},
  author = {Marnie Smith},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.22490},
  eprint = {2607.22490},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.22490},
}
RIS
TY  - UNPB
TI  - The semiclassical limit from Hartree to Vlasov at positive density: strong uniform-in-time convergence and scattering
AU  - Marnie Smith
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2607.22490
UR  - https://arxiv.org/abs/2607.22490
AB  - Strong semiclassical convergence from the Hartree equation to the Vlasov equation is established in three dimensions near Penrose-stable homogeneous steady states at positive density. For sufficiently regular integrable interaction kernels, an O(ℏ^2) convergence rate in weighted Sobolev spaces is proved on finite time intervals. Combining this estimate with earlier uniform-in-ℏ phase-mixing and scattering bounds for the Hartree equation yields global existence and scattering for the corresponding Vlasov solution through the semiclassical limit. If the quantum and classical initial perturbations are aligned, the semiclassical convergence is uniform for all times, and the Wigner transforms of the quantum scattering profiles converge to the classical scattering profile at the explicit rate O((lnℏ^−1)^−1/3).
ER  -