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arXiv · 2610.36240

The Keller problem for the lowest positive eigenvalue of Dirac operators with Lorentz scalar potentials

David Gontier·Hanne Van Den Bosch
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Abstract

We study the Keller problem for the Dirac operator $\mathcal{D}_m - Vβ$ in $\mathbb{R}^d$, 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.

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David Gontier, Hanne Van Den Bosch. 2026-10-26. The Keller problem for the lowest positive eigenvalue of Dirac operators with Lorentz scalar potentials. https://arxiv.org/abs/2610.36240

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