Free harmonic oscillators, Jack polynomials and Calogero-Sutherland systems

Abstract

The algebraic structure and the relationships between the eigenspaces of the Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators. We find an exact connection between the simultaneous non-symmetric eigenfunctions of the $A_{N-1}$ Cherednik operators, from which the eigenfunctions of the CSM and SM are constructed, and the monomials. This construction, not only, allows one to write down a harmonic oscillator algebra involving the Cherednik operators, which yields the raising and lowering operators for both of these models, but also shows the connection of the CSM with free oscillators and the SM with free particles on a circle. We also point out the subtle differences between the excitations of the CSM and the SM.

18 pages, REVTeX, Minor corrections: two equations are corrected; final version

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• Coherent states for exactly solvable potentials
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• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• A Unified Algebraic Approach to Few and Many-Body Correlated Systems
• Duality and quasiparticles in the Calogero-Sutherland model: Some exact results
• Duality between the quantum inverted harmonic oscillator and inverse square potentials
• Dunkl symplectic algebra in generalized Calogero models
• Two-particle Wigner functions in a one-dimensional Calogero-Sutherland potential
• On the Applications of a New Technique to Solve Linear Differential Equations, with and without Source