Coupling and particle number intertwiners in the Calogero model

Abstract

It is long known that quantum Calogero models feature intertwining operators, which increase or decrease the coupling constant by an integer amount, for any fixed number of particles. We name these as ``horizontal'' and construct new ``vertical'' intertwiners, which \emph{change the number of interacting particles} for a fixed but integer value of the coupling constant. The emerging structure of a grid of intertwiners exists only in the algebraically integrable situation (integer coupling) and allows one to obtain each Liouville charge from the free power sum in the particle momenta by iterated intertwining either horizontally or vertically. We present recursion formulæ for the intertwiners as a factorization problem for partial differential operators and prove their existence for small values of particle number and coupling. As a byproduct, a new basis of non-symmetric Liouville integrals appears, algebraically related to the standard symmetric one.

Title change, reference added, note added

References in corpus (15)

• Physics and Mathematics of Calogero particles
• Black Holes and Calogero Models
• Cuboctahedric Higgs oscillator from the Calogero model
• The quantum angular Calogero-Moser model
• Superintegrability of (generalized) Calogero models with oscillator or Coulomb potential
• Nonlinear supersymmetry in the quantum Calogero model
• Generalized Lame operators
• Integrability and separation of variables in Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems
• Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics
• $\mathcal{P}\mathcal{T}$ deformation of angular Calogero models
• $\mathcal{P}\mathcal{T}$ deformation of Calogero-Sutherland models
• Quantum integrability of the deformed elliptic Calogero-Moser problem
• Dunkl symplectic algebra in generalized Calogero models
• Deformed Calogero--Moser operators and ideals of rational Cherednik algebras
• On algebraic integrability of the deformed elliptic Calogero--Moser problem