sl(M+1) Construction of Quasi-solvable Quantum M-body Systems

Details sl(M+1) Construction of Quasi-solvable Quantum M-body Systems sl(M+1) Construction of Quasi-solvable Quantum M-body Systems

2003-06-18

arxiv.org/abs/hep-th/0306174v3

Annals Phys. 309 (2004) 239-280; Erratum-ibid. 320 (2005) 257-259

Physics

High Energy Physics

High Energy Physics - Theory

41 pages, REVTeX4; (v2) published version; (v3) typos corrected

Scientific paper

10.1016/j.aop.2003.08.013

We propose a systematic method to construct quasi-solvable quantum many-body systems having permutation symmetry. By the introduction of elementary symmetric polynomials and suitable choice of a solvable sector, the algebraic structure of sl(M+1) naturally emerges. The procedure to solve the canonical-form condition for the two-body problem is presented in detail. It is shown that the resulting two-body quasi-solvable model can be uniquely generalized to the M-body system for arbitrary M under the consideration of the GL(2,K) symmetry. An intimate relation between quantum solvability and supersymmetry is found. With the aid of the GL(2,K) symmetry, we classify the obtained quasi-solvable quantum many-body systems. It turns out that there are essentially five inequivalent models of Inozemtsev type. Furthermore, we discuss the possibility of including M-body (M>=3) interaction terms without destroying the quasi-solvability.

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