The Keller problem for the lowest positive eigenvalue of Dirac operators with Lorentz scalar potentials
Abstract
We study the Keller problem for the Dirac operator 𝒟m − Vβ in ℝd, and minimize its lowest positive eigenvalue over all potentials V with prescribed Lp norm. We prove that optimal potentials always exist and that the corresponding eigenspinors solve the nonlinear Soler equation. This provides a variational proof of existence of solitary waves for this equation. We prove that there are no zero modes for such operators in dimensions $1$ and $2$, and conjecture that zero mode may exist in dimension $3$. In dimension d = 2, we provide numerical evidence that the optimal potentials concentrate on flower-shape regions, which are not necessarily radial, showing that Faber--Krahn type results may fail for such Dirac operators. In dimension $1$, all quantities are explicit.
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BibTeX
@misc{gontier2026keller,
title = {{The Keller problem for the lowest positive eigenvalue of Dirac operators with Lorentz scalar potentials}},
author = {David Gontier and Hanne Van Den Bosch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36240},
eprint = {2610.36240},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36240},
}
RIS
TY - UNPB
TI - The Keller problem for the lowest positive eigenvalue of Dirac operators with Lorentz scalar potentials
AU - David Gontier
AU - Hanne Van Den Bosch
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36240
UR - https://arxiv.org/abs/2610.36240
AB - We study the Keller problem for the Dirac operator 𝒟_m − Vβ in ℝ^d, and minimize its lowest positive eigenvalue over all potentials V with prescribed L^p norm. We prove that optimal potentials always exist and that the corresponding eigenspinors solve the nonlinear Soler equation. This provides a variational proof of existence of solitary waves for this equation. We prove that there are no zero modes for such operators in dimensions $1$ and $2$, and conjecture that zero mode may exist in dimension $3$. In dimension d = 2, we provide numerical evidence that the optimal potentials concentrate on flower-shape regions, which are not necessarily radial, showing that Faber--Krahn type results may fail for such Dirac operators. In dimension $1$, all quantities are explicit.
ER -