Widom asymptotics for spatial compressions of Schrödinger spectral projections
Abstract
We establish a Widom trace formula for spatial compressions of exact spectral projections of semiclassical Schrödinger operators. The potential is smooth near a compact classically allowed region, and the observation domain lies strictly inside that region. For every continuous spectral function with positive power decay at both endpoints, we obtain a trace formula at scale h1−nlog h−1 whose coefficient separates into a boundary integral and a universal spectral functional. This yields the exact von Neumann and Rényi entanglement asymptotics for every fixed positive order, and proves their universal ratios to the particle-number variance. The proof combines commutator estimates in every fixed positive Schatten class with a local trace formula obtained from a boundary-adapted normal form and a Hardy model. In dimensions n≥2, a positive partition and the known variance asymptotic control the grazing contribution; in one dimension all boundary Fermi points are transverse. A harmonic-oscillator counterexample shows that continuity and weighted integrability alone are insufficient for the trace formula.
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BibTeX
@misc{ma2026widom,
title = {{Widom asymptotics for spatial compressions of Schrödinger spectral projections}},
author = {Pan Ma and Daoyuan Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29076},
eprint = {2610.29076},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29076},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Widom asymptotics for spatial compressions of Schrödinger spectral projections
AU - Pan Ma
AU - Daoyuan Zheng
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29076
UR - https://arxiv.org/abs/2610.29076
AB - We establish a Widom trace formula for spatial compressions of exact spectral projections of semiclassical Schrödinger operators. The potential is smooth near a compact classically allowed region, and the observation domain lies strictly inside that region. For every continuous spectral function with positive power decay at both endpoints, we obtain a trace formula at scale h^1−nlog h^−1 whose coefficient separates into a boundary integral and a universal spectral functional. This yields the exact von Neumann and Rényi entanglement asymptotics for every fixed positive order, and proves their universal ratios to the particle-number variance. The proof combines commutator estimates in every fixed positive Schatten class with a local trace formula obtained from a boundary-adapted normal form and a Hardy model. In dimensions n≥2, a positive partition and the known variance asymptotic control the grazing contribution; in one dimension all boundary Fermi points are transverse. A harmonic-oscillator counterexample shows that continuity and weighted integrability alone are insufficient for the trace formula.
ER -