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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.

对于具有离散能谱的系统，证明了量子谱$ 0 (hbar)$部分与平衡态耦合振荡频率的一般定理。对于Calogero-Sutherland和rujsenaars - schneider -van Diejen系统，该定理得到了广泛和一般的验证。描述精确可解和准精确可解系统平衡点的多项式现在已经被很好地理解了。经典的正交多项式，Hermite， Laguerre和Jacobi多项式及其变形，Meixner-Pollaczek，连续（对偶）Hahn， Wilson和Askey-Wilson多项式都是众所周知的。我们证明了后者描述的是rujsenaars - schneider -van Diejen系统的平衡位置，它是Calogero-Sutherland系统的可积变形。此外，这些变形多项式是具有形状不变势的“离散”单粒子量子力学的精确特征函数。决定这些平衡位置的方程和决定准精确可解系统光谱的方程具有与Bethe ansatz方程非常相似的形式，这为新的研究提供了一个有趣的方向。对于具有最一般椭圆势的精确和准精确可解的多粒子/自旋系统，高自旋Belavin系统和与rujsenaars - schneider系统解相关的椭圆量子群及其Bethe ansatz方程的结构已经取得了实质性进展。