发表机构
Sapienza University of Rome; Beijing Institute of Technology（罗马第一大学; 北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明平面临界电磁哈密顿量的Møller波算子在$L^p$上有界，并确定Aharonov--Bohm模型中Friedrichs和Krein实现下波算子在加权$L^p$空间有界性的精确指数范围，揭示边界条件对可允许指数的影响。
AI 中文摘要
我们研究平面上标度临界电磁哈密顿量的Møller波算子。对于光滑的横向磁场和角向电势，在角算子非负且磁通量位于$\frac12\Z$之外的情况下，我们证明了相对于$-\Delta$的波算子存在，在$L^2$上酉，并且连同其伴随算子，在每一个$L^p$（$1<p<\infty$）上有界。然后我们专门研究Aharonov--Bohm模型，并确定其波算子在加权$L^p(\R^2,|x|^\beta\\,dx)$空间上有界性的精确范围，分别针对Friedrichs和Krein实现。在Friedrichs情形下，这特别给出了已知的所有$L^p$空间（$1<p<\infty$）上的有界性，而在$p=1,\infty$时有界性不成立。在Krein情形下，波算子和伴随算子恰好在$2/(2-\eta_\alpha)<p<2/\eta_\alpha$时有界，其中$\eta_\alpha=\max\{\alpha,1-\alpha\}$（这里$\alpha\in(0,1)$）。因此边界条件改变了允许的指数范围。
英文摘要
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 $\frac12\Z$, we prove that the wave operators relative to $-Δ$ exist, are unitary on $L^2$, and, together with their adjoints, are bounded on every $L^p$, $1<p<\infty$. 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 $1<p<\infty$, while boundedness fails at $p=1,\infty$. 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 $α\in(0,1)$). Thus the boundary condition changes the admissible exponents.
Comments32 pages, Comments are welcome!
