AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models

@article{Finkel2001ANTypeDO,
  title={AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models},
  author={Federico Finkel and David G{\'o}mez-Ullate and A Gonz{\'a}lez-L{\'o}pez and M. A. Rodŕıguez and Renat Z. Zhdanov},
  journal={Communications in Mathematical Physics},
  year={2001},
  volume={221},
  pages={477-497}
}
Abstract: A new family of AN-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians. 
Dunkl Operators and Calogero-Sutherland Models
We describe a general method for constructing (scalar or spin) Calogero-Sutherland models of A n or BC N type, which are either exactly or quasi-exactly solvable. Our approach is based on theExpand
On the Sutherland Spin Model of BN Type and Its Associated Spin Chain
Abstract: The BN hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals ofExpand
Integrable N-particle Hamitnonians with Yangian or reflection algebra symmetry
We use the Dunkl operator approach to construct one-dimensional integrable models describing N particles with internal degrees of freedom. These models are described by a general HamiltonianExpand
Quasi-exact solvability of Inozemtsev models
Finite-dimensional spaces which are invariant under the action of the Hamiltonian of the BCN Inozemtsev model are introduced, and it is shown that the commuting operators of conserved quantities alsoExpand
Quasi-exact solvability in a general polynomial setting
Our goal in this paper is to extend the theory of quasi-exactly solvable Schrodinger operators beyond the Lie-algebraic class. Let be the space of nth degree polynomials in one variable. We firstExpand
Quasi-Exactly Solvable N-Body Spin Hamiltonians with Short-Range Interaction Potentials ?
We review some recent results on quasi-exactly solvable spin models presenting near-neighbors interactions. These systems can be understood as cyclic generalizations of the usual Calogero-SutherlandExpand
On the solutions of 3-particle spin elliptic Calogero–Moser systems
Abstract We describe a class of the singular solutions to the multicomponent analogs of the Lame equation, arising as equations of motion of the elliptic Calogero–Moser systems of particles carryingExpand
The spin Sutherland model of DN type and its associated spin chain
Abstract In this paper we study the su ( m ) spin Sutherland (trigonometric) model of D N type and its related spin chain of Haldane–Shastry type obtained by means of Polychronakos's freezing trick.Expand
CORRIGENDUM: Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser (CM) models. Because of the additional q6 (rational models) or sin 22q (trigonometric models)Expand
Rational quantum integrable systems of DN type with polarized spin reversal operators
We study the spin Calogero model of DN type with polarized spin reversal operators, as well as its associated spin chain of Haldane–Shastry type, both in the antiferromagnetic and ferromagneticExpand
...
1
2
3
...

References

SHOWING 1-10 OF 48 REFERENCES
Exact solutions of a new elliptic Calogero–Sutherland model
Abstract A quantum Hamiltonian describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is introduced. For a discrete set ofExpand
Integration of Quantum Many-Body Problems by Affine Knizhnik-Zamolodchikov Equations
Abstract An equivalence of the affine analogues of Knizhnik-Zamolodchikov and Dunkl equations is established for arbitrary root systems. As a corollary, an isomorphism is obtained from the solutionsExpand
Loop Groups, Anyons and the Calogero–Sutherland Model
Abstract:The positive energy representations of the loop group of U(1) are used to construct a boson-anyon correspondence. We compute all the correlation functions of our anyon fields and study anExpand
Hamiltonian systems of Calogero-type, and two-dimensional Yang-Mills theory
We obtain integral representations for the wave functions of Calogerotype systems, corresponding to the finite-dimensional Lie algebras, using exact evaluation of path integral. We generalize theseExpand
The Calogero-Sutherland Model and Generalized Classical Polynomials
Abstract:Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schrödinger operators forExpand
Generalized Calogero models through reductions by discrete symmetries
Abstract We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a classical Calogero model by a subset of its discrete symmetries. Such reductions reproduceExpand
Spin-dependent extension of Calogero-Sutherland model through anyon-like representations of permutation operators
We consider a AN−1 type of spin-dependent Calogero-Sutherland model, containing an arbitrary representation of the permutation operators on the combined internal space of all particles, and find thatExpand
The orthogonal eigenbasis and norms of eigenvectors in the spin Calogero - Sutherland model
Using a technique based on the Yangian Gelfand - Zetlin algebra and the associated Yangian Gelfand - Zetlin bases we construct an orthogonal basis of eigenvectors in the Calogero - Sutherland modelExpand
Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators
We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi-exactly solvable Schrödinger operator onExpand
Quasi-exactly-solvable problems andsl(2) algebra
Recently discovered quasi-exactly-solvable problems of quantum mechanics are shown to be related to the existence of the finite-dimensional representations of the groupSL(2,Q), whereQ=R, C. It isExpand
...
1
2
3
4
5
...