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

摘要

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 greater than or equal to 3) interaction terms without destroying the quasi-solvability. (C) 2003 Elsevier Inc. All rights reserved. [References: 69]