论文标题
双曲线kac-moody calogero型号
A hyperbolic Kac-Moody Calogero model
论文作者
论文摘要
基于双曲线kac-moody代数提出了一种新型的量子Calogero模型。我们在最简单的级别3屈曲谎言代数$ ae_3 $的Minkowskian根空间上制定非层次量子力学，其实际根具有反平方的电位，并将其减少到单元的未来双曲线。通过立体投影，这定义了具有独特潜力的庞加莱磁盘上的量子力学。由于$ ae_3 $的Weyl oft of $ \ mathbb {z} _2 $ psl的扩展（2，$ \ m m mathbb {z} _2 $），因此该模型自然是在复杂的上半平面上配制的，其电位是一个真正的模块化函数。我们介绍并说明了$ AE_3 $的相关功能，给潜在的一些近似值，并将其重写为（几乎在汇聚）庞加莱系列。在Minkowski空间和双曲体上构建并研究了标准的DUNKL操作员。在前一种情况下，发现它们的交通量受到双曲线类型的等级-2亚组的阻碍（斐波那契序列给出的最简单的类型），对模型的整合性产生了怀疑。 Don Zagier的附录调查了电势的可计算性。我们预见到宇宙学和量子混乱的应用。
A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra $AE_3$ with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of $AE_3$ is a $\mathbb{Z}_2$ extension of the modular group PSL(2,$\mathbb{Z}_2$), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of $AE_3$, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.
