Aharonov-Bohm型磁Schroedinger算子的节集与能量最小划分
Nodal sets of magnetic Schroedinger operators of Aharonov-Bohm type and energy minimizing partitions
摘要：
本文考虑了在Aharonov-Bohm型半整数环磁场存在下平面上的定常Schroedinger算子。我们分析了一类解的节点区域，使得节点集由连接奇点与边界的规则弧组成。在一个磁极的情况下，这是自由移动的域，节线可以集群解剖域的三个部分。我们的主要结果表明，磁能是临界的（相对于磁极），当且仅当这样的配置发生。此外，节点区域形成域的最小3-分区（相对于与方程相关联的实能量），配置是唯一的，并且连续地依赖于数据。所进行的分析与[20]中引入的谱最小划分的概念有关。由于它涉及本征函数，我们同样表明，临界点的瑞利商对应于多个集群的节线。
英文摘要：
In this paper we consider a stationary Schroedinger operator in the plane, in presence of a magnetic field of Aharonov-Bohm type with semi-integer circulation. We analyze the nodal regions for a class of solutions such that the nodal set consists of regular arcs, connecting the singular points with the boundary. In case of one magnetic pole, which is free to move in the domain, the nodal lines may cluster dissecting the domain in three parts. Our main result states that the magnetic energy is critical (with respect to the magnetic pole) if and only if such a configuration occurs. Moreover the nodal regions form a minimal 3-partition of the domain (with respect to the real energy associated to the equation), the configuration is unique and depends continuously on the data. The analysis performed is related to the notion of spectral minimal partition introduced in [20]. As it concerns eigenfunctions, we similarly show that critical points of the Rayleigh quotient correspond to multiple clustering of the nodal lines.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
