sl(M+1) construction of quasi-solvable quantum M-body systems

Abstract 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.