Hidden algebra of the $N$-body Calogero problem

Abstract

A certain generalization of the algebra $gl(N,\mathbf{R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\mathbf{R}}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.

10pp., CWRU-Math, October 1993

References in corpus (3)

• Exchange Operator Formalism for Integrable Systems of Particles
• Explicit Solution to the N-Body Calogero Problem
• The Calogero Model - Anyonic Representation, Fermionic Extension and Supersymmetry

Cited by in corpus (10)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• On Dunkl angular momenta algebra
• Cat-States in the Framework of Wigner-Heisenberg Algebra
• Quasi-Exact Solvability and the direct approach to invariant subspaces
• Dunkl symplectic algebra in generalized Calogero models
• Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems
• Calogero-Vasiliev Oscillator in Dynamically Evolving Curved Spacetime
• On symmetries of generalized Calogero model and Polychronakos-Frahm chain
• Calogero-like model without rearrangement symmetry
• Dynamical symmetries of the Calogero-Coulomb model