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

对于具有离散能谱的系统，证明了量子谱的$o(Hbar)$部分与平衡点的耦合振荡频率有关的一般定理。该定理在Calogero-Sutherland和Ruijsenaars-Schneider-van Diejen系统中得到了广泛和普遍的证明。描述精确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多项式是众所周知的。我们证明了后者描述的是Ruijsenaars-Schneider-van Diejen系统的平衡位置，它是Calogero-Sutherland系统的可积变形。此外，这些变形的多项式是具有形状不变势的离散单粒子量子力学的精确本征函数。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，这表明了一个有趣的新研究方向。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，与Ruijsenaars-Schneider系统的解有关的高自旋Belavin系统和椭圆量子群及其Bethe ansatz方程的结构已经取得了实质性的进展。

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.