Nonlinear supersymmetry in the quantum Calogero model

Abstract

It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g+1. For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order n(n-1)(2g-1)/2 in the momenta and squares to a polynomial in the Liouville charges. With a natural Z_2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.

1+21 pages; v2: minor corrections, typos in eq.(5.18) fixed, 3 refs. added, published version

References in corpus (7)

• Physics and Mathematics of Calogero particles
• Self-isospectrality, special supersymmetry, and their effect on the band structure
• Aharonov-Bohm effect on AdS_2 and nonlinear supersymmetry of reflectionless Poschl-Teller system
• Spectral singularities in PT-symmetric periodic finite-gap systems
• PT-symmetric deformations of integrable models
• Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems
• Some Orthogonal Polynomials in Four Variables

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• Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis
• The Condensation of Ideal Dunkl-Bose Gas in Power-Law Traps
• The structure of invariants in conformal mechanics
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• A Path Integral Treatment of Time-dependent Dunkl Quantum Mechanics
• One-dimensional Dunkl Quantum Mechanics: A Path Integral Approach
• Hidden symmetry and (super)conformal mechanics in a monopole background
• $\mathcal{P}\mathcal{T}$ deformation of angular Calogero models
• Integrability, intertwiners and non-linear algebras in Calogero models
• $\mathcal{P}\mathcal{T}$ deformation of Calogero-Sutherland models
• Time-dependent Dunkl-Pauli Oscillator
• Dunkl-Klein-Gordon Equation in Higher Dimensions
• Nonlinear supersymmetry as a hidden symmetry
• Dynamical symmetries of the Calogero-Coulomb model
• Hidden symmetries and nonlinear (super)algebras
• Algebraic integrability of $\mathcal{P}\mathcal{T}$-deformed Calogero models
• Coupling and particle number intertwiners in the Calogero model