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A long time Gutzwiller trace formula for perturbations of Schrödinger operators near Anosov energy levels
Abstract
Let Ph be a semiclassical pseudodifferential operator on a closed Riemannian manifold (M,g), with principal symbol p. We consider perturbations Ph+hαQ, with 0<α≤ 1, where Q is a semiclassical pseudodifferential operator whose symbol oscillates at scale hβ, with 2β<α. Assuming that the classical dynamics of the unperturbed system on a regular energy surface p−1(E) is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order |log h|. This formula relates the distribution of energy levels of the quantum system near the energy E to the periodic orbits of its classical counterpart. The perturbation hαQ affects only the phases appearing in the trace formula.
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BibTeX
@misc{moy2026long,
title = {{A long time Gutzwiller trace formula for perturbations of Schrödinger operators near Anosov energy levels}},
author = {Julien Moy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09046},
eprint = {2610.09046},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09046},
}
RIS
TY - UNPB
TI - A long time Gutzwiller trace formula for perturbations of Schrödinger operators near Anosov energy levels
AU - Julien Moy
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09046
UR - https://arxiv.org/abs/2610.09046
AB - Let P_h be a semiclassical pseudodifferential operator on a closed Riemannian manifold (M,g), with principal symbol p. We consider perturbations P_h+h^αQ, with 0<α≤ 1, where Q is a semiclassical pseudodifferential operator whose symbol oscillates at scale h^β, with 2β<α. Assuming that the classical dynamics of the unperturbed system on a regular energy surface p^−1(E) is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order |log h|. This formula relates the distribution of energy levels of the quantum system near the energy E to the periodic orbits of its classical counterpart. The perturbation h^αQ affects only the phases appearing in the trace formula.
ER -