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arXiv 2201.03610math-phmath.APmath.MP
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Dirac方程零模的一个精确判据

A sharp criterion for zero modes of the Dirac equation

Rupert L. Frank, Michael Loss

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摘要：

证明了$\Vert A \Vert_{L^d}^2 \ge \frac{d}{d-2}\, S_d$是$d$维Dirac方程$γ\cdot (-i\nabla -A)ψ= 0$存在非平凡解的必要条件.这里，$S_d$是尖锐的Sobolev常数。如果$d$是奇数，而$\Vert A \Vert_{L^d}^2= \frac{d}{d-2}\, S_d$是奇数，则存在允许零模的矢量势。给出了这些矢势及其相应零模的完整分类。

英文摘要：

It is shown that $\Vert A \Vert_{L^d}^2 \ge \frac{d}{d-2}\, S_d$ is a necessary condition for the existence of a nontrivial solution of the Dirac equation $γ\cdot (-i\nabla -A)ψ= 0$ in $d$ dimensions. Here, $S_d$ is the sharp Sobolev constant. If $d$ is odd and $\Vert A \Vert_{L^d}^2= \frac{d}{d-2}\, S_d$, then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.

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