Solvable scalar and spin models with near-neighbors interactions

TL;DR

This paper introduces new exactly solvable scalar and spin models with near-neighbor interactions using an extension of Dunkl operators, revealing finite and infinite spectra and explicit eigenfunctions.

Contribution

It extends Dunkl operator formalism to construct new solvable spin models with near-neighbor interactions, providing explicit eigenfunctions and spectral properties.

Findings

Trigonometric models have finite energy levels in the center of mass frame.

Rational models exhibit an equally spaced infinite algebraic spectrum.

Explicit eigenfunctions are computed for selected models, involving Laguerre and Jacobi polynomials.

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.