论文标题
具有复杂磁场
Cylindrical first order superintegrability with complex magnetic fields
论文作者
论文摘要
本文是对量子力学中三维欧几里得空间$ \ mathbb {e} _3 $在量子力学中的磁场上进行磁场上的可整合哈密顿系统的研究的贡献。与实验证实其物理相关性后，数学界对复杂电磁场的兴趣日益增强相反[X. Peng等人，物理。莱特牧师。 114（2015）]，到目前为止，他们在不断增长的有关可累加性的文献中还没有解决。在这里，我们通过寻找圆柱类型集成系统的其他运动积分来冒险进入该领域。我们发现，已经知道的系统可以通过接纳复杂的耦合常数扩展到该领域。除了它们外，我们还发现了一个新系统，其运动积分还具有复杂的常数。所有这些系统都是多院的。由于非富裕设置和损失的量规不变性，对这些系统的严格数学分析是具有挑战性的。我们正式进行，并提出解决这些问题的解决方案，作为一个公开挑战。
This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space $\mathbb{E}_3$ in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [X. Peng et al., Phys. Rev. Lett. 114 (2015)], they were so far not addressed in the growing literature on superintegrability. Here we venture into this field by searching for additional first order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.
