Quantum Calogero-Moser models: integrability for all root systems
Quantum Calogero-Moser models: integrability for all root systems
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Quantum Calogero-Moser 模型：所有根系统的可集成性
DOI：
10.1088/0305-4470/33/49/303
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
R. Sasaki
中科院分区：
文献类型：
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作者：
S. Khastgir;A. Pocklington;R. Sasaki
讨论了基于任意根系的量子Calogero-Moser模型的可积性问题。对于退化势的模型，即有/无谐波约束力的有理模型，双曲模型和三角模型，我们证明了以下所有的根系。(i)用Lax矩阵L的总和构造量子守恒量的完备集，即Sigma（μ，nu是R的一个元素）（L-n）（μ nu），其中R是在Coxeter群作用下不变的R-r向量集.它们形成一个单一的考克斯特轨道。(ii)Liouville可积性的证明(iii)量子哈密顿量的三角性和整个离散谱。广义杰克多项式被定义为唯一的本征函数的所有根系统的哈密尔顿。(iv)Lax算子和Dunkl算子的等价性。(v)所有激发态的创造算符代数构造。这些结果主要是对基于A级数（即su（N）型）根系的模型的已知结果的推广。
The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems. (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix L, i.e. Sigma (mu,nu is an element ofR)(L-n)(mu nu), in which R is a set of R-r vectors invariant under the action of the Coxeter group. They form a single Coxeter orbit. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalized Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalizations of the results known for the models based on the A series, i.e. su(N)-type, root systems.