Coxeter群的反线性变形及其在Calogero模型中的应用
Antilinear deformations of Coxeter groups, an application to Calogero models
摘要：
我们构造复根空间保持不变的反线性对合下的所有Coxeter群。我们提供了两种可供选择的结构：一种是基于变形的因素的Coxeter元素和其他的基础上变形的最长元素的Coxeter群。由于承认反线性对称性的非厄米哈密顿算子可以用来定义具有实离散能谱的一致量子力学系统，我们随后采用我们的构造来制定在这些扩展的Coxeter群下保持不变的Coxeter模型的变形。我们提供明确的和通用的解决方案，这些模型的薛定谔方程的本征能量和相应的波函数。这些新的模型的一个新的特点是，当与未变形的情况下，他们的解决方案通常不再是奇异的交换量的粒子小于根的表示空间的维度。模型中所有粒子的同时散射导致了在未变形情况下没有类似过程的任意子交换因子。
英文摘要：
We construct complex root spaces remaining invariant under antilinear involutions related to all Coxeter groups. We provide two alternative constructions: One is based on deformations of factors of the Coxeter element and the other based on the deformation of the longest element of the Coxeter group. Motivated by the fact that non-Hermitian Hamiltonians admitting an antilinear symmetry may be used to define consistent quantum mechanical systems with real discrete energy spectra, we subsequently employ our constructions to formulate deformations of Coxeter models remaining invariant under these extended Coxeter groups. We provide explicit and generic solutions for the Schroedinger equation of these models for the eigenenergies and corresponding wavefunctions. A new feature of these novel models is that when compared with the undeformed case their solutions are usually no longer singular for an exchange of an amount of particles less than the dimension of the representation space of the roots. The simultaneous scattering of all particles in the model leads to anyonic exchange factors for processes which have no analogue in the undeformed case.
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