Solvable scalar and spin models with near-neighbors interactions

Abstract

We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass frame, while the rational models are shown to possess an equally spaced infinite algebraic spectrum. For the trigonometric and one of the rational models, the corresponding eigenfunctions are explicitly computed. We also study the scalar reductions of the models, some of which had already appeared in the literature, and compute their algebraic eigenfunctions in closed form. In the rational cases, for which only partial results were available, we give concise expressions of the eigenfunctions in terms of generalized Laguerre and Jacobi polynomials.

10 pages, LaTeX. Minor typos corrected

References in corpus (27)

• Exchange Operator Formalism for Integrable Systems of Particles
• The Calogero-Sutherland Model and Generalized Classical Polynomials
• Exact operator solution of the Calogero-Sutherland model
• Lattice Integrable Systems of Haldane-Shastry Type
• Exact Dynamical Correlation Functions of Calogero-Sutherland Model and One-Dimensional Fractional Statistics
• Hamiltonian systems of Calogero type and two dimensional Yang-Mills theory
• Fractional Statistics in One-Dimension: View From An Exactly Solvable Model
• Calogero-Moser Systems in SU(N) Seiberg-Witten Theory
• Exact Spectrum of SU(n) Spin Chain with Inverse-Square Exchange
• Hidden Algebras of the (super) Calogero and Sutherland models
• Waves and Solitons in the Continuum Limit of the Calogero-Sutherland Model
• Explicit Relation of Quantum Hall Effect and Calogero-Sutherland Model
• Exact Solution of Long-Range Interacting Spin Chains with Boundaries
• Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems
• An exactly solvable many-body problem in one dimension
• A class of N-body problems with nearest- and next-to-nearest neighbour interactions
• Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials
• Multicomponent Calogero Model of $B_N$-Type Confined in Harmonic Potential
• A_N-type Dunkl operators and new spin Calogero-Sutherland models
• Integrable $1/r^2$ Spin Chain with Reflecting End
• New spin Calogero-Sutherland models related to B_N-type Dunkl operators
• Super-Calogero-Moser-Sutherland systems and free super-oscillators : a mapping
• Random Matrix Model and the Calogero-Sutherland Model: A Novel Current-Density Mapping
• Loop groups, anyons and the Calogero-Sutherland model
• Spin dependent extension of Calogero-Sutherland model through anyon like representations of permutation operators
• Non-Hermiticity and Universality
• Spin chains from super-models

Cited by in corpus (8)

• Physics and Mathematics of Calogero particles
• Exact ground states of quantum many-body systems under confinement
• Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
• Exchange operator formalism for N-body spin models with near-neighbors interactions
• A novel quasi-exactly solvable spin chain with nearest-neighbors interactions
• Taxonomy of integrable and ground-state solvable models: Jastrow wave functions on graphs and parent Hamiltonians
• Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions
• Quasi-Exactly Solvable N-Body Spin Hamiltonians with Short-Range Interaction Potentials