A Quasi-Exactly Solvable N-Body Problem with the sl(N+1) Algebraic Structure

TL;DR

This paper constructs a new N-body quantum system with a deformed Calogero model structure, exhibiting quasi-exact solvability and algebraic features related to sl(N+1), extending the understanding of integrable many-body systems.

Contribution

It introduces a novel N-body Hamiltonian with sl(N+1) algebraic structure, derived from a one-particle quasi-exactly solvable system, and demonstrates its quasi-exact solvability and symmetry properties.

Findings

The Hamiltonian reduces to a quadratic combination of sl(N+1) generators.

The interaction potential includes two-body and force-center terms.

The system is quasi-exactly solvable for specific cohomology parameter values.

Abstract

Starting from a one-particle quasi-exactly solvable system, which is characterized by an intrinsic sl(2) algebraic structure and the energy-reflection symmetry, we construct a daughter N-body Hamiltonian presenting a deformation of the Calogero model. The features of this Hamiltonian are (i) it reduces to a quadratic combination of the generators of sl(N+1); (ii) the interaction potential contains two-body terms and interaction with the force center at the origin; (iii) for quantized values of a certain cohomology parameter n it is quasi-exactly solvable, the multiplicity of states in the algebraic sector is (N+n)!/(N!n!); (iv) the energy-reflection symmetry of the parent system is preserved.