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Dynamical symmetries of the Calogero-Coulomb model

Tigran Hakobyan

Abstract

We construct the dynamical symmetry of the quantum Calogero model with particle exchange in a confining Coulomb field. This symmetry is governed by the algebra $so(N+1,2)$, deformed by exchange (Dunkl) operators, with its invariant sector generated by the Dunkl angular momentum tensor and the modified Laplace-Runge-Lenz vector. The equidistant analogue of the Hamiltonian, with a linear spectrum, is expressed in terms of the conformal subalgebra $so(1,2)$. In addition, the wave functions of the Calogero-Coulomb Hamiltonian are classified into infinite-dimensional lowest-weight $so(1,2)$ multiplets.

Dynamical symmetries of the Calogero-Coulomb model

Abstract

We construct the dynamical symmetry of the quantum Calogero model with particle exchange in a confining Coulomb field. This symmetry is governed by the algebra , deformed by exchange (Dunkl) operators, with its invariant sector generated by the Dunkl angular momentum tensor and the modified Laplace-Runge-Lenz vector. The equidistant analogue of the Hamiltonian, with a linear spectrum, is expressed in terms of the conformal subalgebra . In addition, the wave functions of the Calogero-Coulomb Hamiltonian are classified into infinite-dimensional lowest-weight multiplets.
Paper Structure (16 sections, 95 equations)

This paper contains 16 sections, 95 equations.

Table of Contents

  1. Introduction
  2. Calogero--Coulomb system
  3. Hamiltonian via exchange (Dunkl) operators
  4. Symmetries of Calogero--Coulomb model
  5. Conformal algebra and the equidistant Calogero--Coulomb system
  6. Three-dimensional conformal algebra
  7. Equidistant analogue of Calogero--Coulomb Hamitonian
  8. Dynamical symmetry of the equidistant Calogero--Coulomb system
  9. Dynamical symmetry algebra for the parameter $E=-1/2$
  10. Dynamical symmetry for general $E$
  11. $SO(1,2)$ structure of the Calogero--Coulomb wave functions
  12. Stationary wave functions in deformed spherical harmonics
  13. Conformal algebra structure of eigenfunctions
  14. Conclusion
  15. Commutation relations in the $\mathcal{H}_gso(2,N)$ algebra
  16. ...and 1 more sections