Quantum versus classical integrability in Calogero-Moser systems
Quantum versus classical integrability in Calogero-Moser systems
复制标题
Calogero-Moser 系统中的量子可积性与经典可积性
DOI：
10.1088/0305-4470/35/33/306
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
Ryu Sasaki
中科院分区：
文献类型：
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作者：
Edward Corrigan;Ryu Sasaki
Calogero-Moser系统是经典的和量子可积的多粒子动力学，定义为任何根系统Δ。具有1/q2势和限制q2势的量子Calogero系统和具有1/sin2q势的Sutherland系统具有由根系统Δ表征的‘整数’能谱。用解析和数值方法研究了相应的经典系统在势的平衡点上的各种量，如最小能量、小振动频率、经典Lax对矩阵的本征值等。令我们惊讶的是，这些经典数据中的大多数也是“整数”，或者说它们似乎是“量化的”。更准确地说，这些量是耦合常数(S)的整数系数多项式。Calogero-Moser系统的量子可积性与经典可积性之间的密切关系值得进行更全面的分析处理，这将有助于更好地理解这些系统和一般的可积系统。
Calogero–Moser systems are classical and quantum integrable multiparticle dynamics defined for any root system Δ. The quantum Calogero systems having 1/q2 potential and a confining q2 potential and the Sutherland systems with 1/sin2q potentials have 'integer' energy spectra characterized by the root system Δ. Various quantities of the corresponding classical systems, e.g. minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices etc, at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also 'integers', or they appear to be 'quantized'. To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero–Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general.