New spin Calogero-Sutherland models related to B_N-type Dunkl operators

Abstract

We construct several new families of exactly and quasi-exactly solvable BC_N-type Calogero-Sutherland models with internal degrees of freedom. Our approach is based on the introduction of two new families of Dunkl operators of B_N type which, together with the original B_N-type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of quadratic combinations involving these three sets of Dunkl operators always yields a spin Calogero-Sutherland model, which is (quasi-)exactly solvable by construction. We show that all the spin Calogero-Sutherland models obtainable within this framework can be expressed in a unified way in terms of a Weierstrass P function with suitable half-periods. This provides a natural spin counterpart of the well-known general formula for a scalar completely integrable potential of BC_N type due to Olshanetsky and Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-exactly solvable potential for two and three particles of spin 1/2.

18 pages, typeset in LaTeX 2e using revtex 4.0b5 and the amslatex package Minor changes in content, one reference added

References in corpus (1)

• A_N-type Dunkl operators and new spin Calogero-Sutherland models

Cited by in corpus (24)

• Physics and Mathematics of Calogero particles
• Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
• Haldane-Shastry spin chains of BC_N type
• The Polychronakos-Frahm spin chain of BC_N type and Berry-Tabor's conjecture
• Exactly solvable D_N-type quantum spin models with long-range interaction
• Calogero-Moser models with noncommutative spin interactions
• On $q$-deformations of the Heun equation
• On the Sutherland Spin Model of B_N Type and its Associated Spin Chain
• On R-matrix valued Lax pairs for Calogero-Moser models
• Source identity and kernel functions for elliptic Calogero-Sutherland type systems
• sl(M+1) Construction of Quasi-solvable Quantum M-body Systems
• Integrable N-particle Hamiltonians with Yangian or Reflection Algebra Symmetry
• Solvable scalar and spin models with near-neighbors interactions
• Twisted spin Sutherland models from quantum Hamiltonian reduction
• Exchange operator formalism for N-body spin models with near-neighbors interactions
• Heun equation and Inozemtsev models
• A novel quasi-exactly solvable spin chain with nearest-neighbors interactions
• Rational quantum integrable systems of D_N type with polarized spin reversal operators
• A novel class of translationally invariant spin chains with long-range interactions
• A Haldane-Shastry spin chain of BC_N type in a constant magnetic field
• On the Inozemtsev model
• Quasi-Exactly Solvable N-Body Spin Hamiltonians with Short-Range Interaction Potentials
• Quasi-exact solvability of Inozemtsev models
• Towards Finite-Gap Integration of the Inozemtsev Model