E. Emsiz

Periodic Integrable Systems with Delta-Potentials

In this paper we study root system generalizations of the quantum Bose-gas on the circle with pair-wise delta-function interactions. The underlying symmetry structures are shown to be governed by the associated graded algebra of Cheredniks (suitably filtered) degenerate double affine Hecke algebra, acting by Dunkl-type differential-reflection operators. We use Gutkins generalization of the equivalence between the impenetrable Bose-gas and the free Fermi-gas to derive the Bethe ansatz equations and the Bethe ansatz eigenfunctions.

Trigonometric Cherednik algebra at critical level and quantum many-body problems

Abstract.For any module over the affine Weyl group we construct a representation of the associated trigonometric Cherednik algebra A(k) at critical level in terms of Dunkl type operators. Under this representation the center of A(k) produces quantum conserved integrals for root system generalizations of quantum spin-particle systems on the circle with delta function interactions. This enables us to translate the spectral problem of such a quantum spin-particle system to questions in the representation theory of A(k). We use this approach to derive the associated Bethe ansatz equations. They are expressed in terms of the normalized intertwiners of A(k).

Spectrum and Eigenfunctions of the Lattice Hyperbolic Ruijsenaars–Schneider System with Exponential Morse Term

We place the hyperbolic quantum Ruijsenaars–Schneider system with an exponential Morse term on a lattice and diagonalize the resulting n-particle model by means of multivariate continuous dual q-Hahn polynomials that arise as a parameter reduction of the Macdonald–Koornwinder polynomials. This allows to compute the n-particle scattering operator, to identify the bispectral dual system, and to confirm the quantum integrability in a Hilbert space setup.

The Semi-Infinite q-Boson System with Boundary Interaction

Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald’s hyperoctahedral Hall–Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices.

Orthogonality of Bethe Ansatz Eigenfunctions for the Laplacian on a Hyperoctahedral Weyl Alcove

We prove the orthogonality of the Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove with repulsive homogeneous Robin boundary conditions at the walls. To this end these eigenfunctions are retrieved as the continuum limit of an orthogonal basis of algebraic Bethe Ansatz eigenfunctions for a finite

Orthogonality of Macdonald polynomials with unitary parameters

{q}

A Generalized Macdonald Operator

q-boson system endowed with diagonal open-end boundary interactions.

Difference equation for the Heckman–Opdam hypergeometric function and its confluent Whittaker limit☆

Abstract We present an explicit branching formula for the six-parameter Macdonald–Koornwinder polynomials with hyperoctahedral symmetry.