Lp bounds for wave operators with critical electromagnetic potentials
Abstract
We study the Møller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside 1/2Z, we prove that the wave operators relative to −Δ exist, are unitary on L2, and, together with their adjoints, are bounded on every Lp, 1<p<∞. We then specialize to the the Aharonov--Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted Lp(R2,|x|β dx) spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all Lp spaces 1<p<∞, while boundedness fails at p=1,∞. In the Krein case, both wave operators and adjoints are bounded precisely when 2/(2−ηα)<p<2/ηα, where ηα=max{α,1−α} (here α∈(0,1)). Thus the boundary condition changes the admissible exponents.
Related
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BibTeX
@misc{dancona2026bounds,
title = {{$L^p$ bounds for wave operators with critical electromagnetic potentials}},
author = {Piero D'Ancona and Xitao Gao and Junyong Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07464},
eprint = {2609.07464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07464},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - L^p bounds for wave operators with critical electromagnetic potentials
AU - Piero D'Ancona
AU - Xitao Gao
AU - Junyong Zhang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07464
UR - https://arxiv.org/abs/2609.07464
AB - We study the Møller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside 1/2Z, we prove that the wave operators relative to −Δ exist, are unitary on L^2, and, together with their adjoints, are bounded on every L^p, 1<p<∞. We then specialize to the the Aharonov--Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted L^p(R^2,|x|^β dx) spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all L^p spaces 1<p<∞, while boundedness fails at p=1,∞. In the Krein case, both wave operators and adjoints are bounded precisely when 2/(2−η_α)<p<2/η_α, where η_α=max{α,1−α} (here α∈(0,1)). Thus the boundary condition changes the admissible exponents.
ER -