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

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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.