PreprintCC BY 4.0HAL
Long-time dynamics for the magnetic Schrödinger equation on the torus
Abstract
We investigate time-dependent semiclassical measures associated with the magnetic Schr\"odinger flow on the flat two-dimensional torus in long-time regimes. Under a geometric nonvanishing assumption on an effective magnetic force, we identify two thresholds and prove a dynamical form of quantum unique ergodicity for such systems. If the time scale τh ≫ h−1/2, then the configuration marginal is the normalized Lebesgue measure on 𝕋2. In addition, if τh ≫ h−1, then the momentum marginal is, conditionally on each regular level set, the normalized invariant measure on that curve.
Related
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BibTeX
@misc{lechaux2026longtime,
title = {{Long-time dynamics for the magnetic Schrödinger equation on the torus}},
author = {Julien Lechaux},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19224},
eprint = {2609.19224},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05750137v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Long-time dynamics for the magnetic Schrödinger equation on the torus
AU - Julien Lechaux
PY - 2026
DA - 2026/09/02
PB - HAL
DO - 10.48550/arxiv.2609.19224
UR - https://hal.science/hal-05750137v1
AB - We investigate time-dependent semiclassical measures associated with the magnetic Schr\"odinger flow on the flat two-dimensional torus in long-time regimes. Under a geometric nonvanishing assumption on an effective magnetic force, we identify two thresholds and prove a dynamical form of quantum unique ergodicity for such systems. If the time scale τ_h ≫ h^−1/2, then the configuration marginal is the normalized Lebesgue measure on 𝕋^2. In addition, if τ_h ≫ h^−1, then the momentum marginal is, conditionally on each regular level set, the normalized invariant measure on that curve.
ER -