Exactly and quasi-Exactly Solvable multi-particle Quantum Systems and Generalized Supersymmetry

精确和准精确可解的多粒子量子系统和广义超对称性

基本信息

批准号：
14540259
负责人：
SASAKI Ryu
金额：
$ 1.79万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

A general theorem relating the $o(hbar)$ part of the quantum spectrum to the frequencies of coupled oscillations at the equilibrium is proved for the systems having discrete energy spectrum. The theorem is verified broadly and generally for the Calogero-Sutherland and Ruijsenaars-Schneider-van Diejen systems. The polynomials describing the equilibrium points of exactly and quasi-exactly solvable systems are now quite well understood. The classical orthogonal polynomials, the Hermite, Laguerre, and Jacobi polynomials and their deformation, the Meixner-Pollaczek, continuous (dual) Hahn, Wilson, and Askey-Wilson polynomials are well-known. We have proved that the latter are describing the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems, which are integrable deformation of the Calogero-Sutherland systems. Moreover, these deformed polynomials are the exact eigenfunctions of the 'discrete' single particle quantum mechanics with shape-invariant potentials. The equations determining these equilibrium positions and those determining the spectra of quasi-exactly solvable systems have a form very similar to the Bethe ansatz equation, which suggests an interesting direction of new research. As for exactly and quasi-exactly solvable multiparticle/spin systems with the most general elliptic potentials, the high-spin Belavin systems and the elliptic quantum groups associated with the solutions of the Ruijsenaars-Schneider systems together with the structure of their Bethe ansatz equations have seen substantial progress.

事实证明，将量子频谱的$ O（HBAR）$一部分与平衡处耦合振荡的频率有关的一般定理证明了具有离散能量谱的系统。该定理对Calogero-Sutherland和Ruijsenaars-Schneider-van Diejen Systems进行了广泛的验证。现在已经充分理解了描述准确和准确解决系统的平衡点的多项式。经典的正交多项式，Hermite，Laguerre和Jacobi多项式及其变形，Meixner-Pollaczek，连续（Dual）Hahn，Wilson和Askey-Wilson的多项式。我们已经证明了后者描述了Ruijsenaars-Schneider-Van diejen Systems的平衡位置，该系统是Calogero-Sutherland Systems的可整合变形。此外，这些变形的多项式是具有形状不变电位的“离散”单粒子量子力学的确切特征函数。确定这些均衡位置的方程式以及确定准实心可解析系统光谱的方程式具有与伯特·安萨兹方程非常相似的形式，这表明了新研究的有趣方向。至于具有最一般椭圆电位的准确和准溶解的多粒子/自旋系统，高旋转的Belavin系统以及与Ruijsenaars-Schneider Systems溶液相关的椭圆量子组以及其Bethe Ansatz方程的结构，已经看到了实质性的进步。