arXiv subjects
Publications and source records attributed to E. Emsiz.
We construct the basic representation of the double affine Hecke algebra at critical level $q=1$ associated to an irreducible reduced affine root system $R$ with a reduced gradient root system. For $R$ of untwisted type such a representation was studied by Oblomkov [O04] and further detailed by Gehles [G06] in the presence of minuscule weights.
We show that hyperoctahedral Whittaker functions---diagonalizing an open quantum Toda chain with one-sided boundary potentials of Morse type---satisfy a dual system of difference equations in the spectral variable. This extends a well-known bispectral duality between the nonperturbed open quantum Toda chain and a strong-coupling limit of the rational Macdonald-Ruijsenaars difference operators. It is manifest from the difference equations in question that the hyperoctahedral Whittaker function is entire as a function of the spectral variable.
Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey-Wilson, Wilson and continuous Hahn families.
We employ a multivariate extension of the Gauss quadrature formula, originally due to Berens, Schmid and Xu [BSX95], so as to derive cubature rules for the integration of symmetric functions over hypercubes (or infinite limiting degenerations thereof) with respect to the densities of unitary random matrix ensembles. Our main application concerns the explicit implementation of a class of cubature rules associated with the Bernstein-Szeg\"o polynomials, which permit the exact integration of symmetric rational functions with prescribed poles at coordinate hyperplanes against unitary circular Jacobi distributions stemming from the Haar measures on the symplectic and the orthogonal groups.
We glue two families of Bernstein-Szego polynomials to construct the eigenbasis of an associated finite-dimensional Jacobi matrix. This gives rise to finite orthogonality relations for this composite eigenbasis of Bernstein-Szego polynomials. As an application, a number of Gauss-like quadrature rules are derived for the exact integration of rational functions with prescribed poles against the Chebyshev weight functions.
We prove the Plancherel formula for a four-parameter family of discrete Fourier transforms and their multivariate generalizations stemming from corresponding generalized Schur polynomials. For special choices of the parameters, this recovers the sixteen classic discrete sine- and cosine transforms DST-1,...,DST-8 and DCT-1,...,DCT-8, as well as recently studied (anti-)symmetric multivariate generalizations thereof.
We employ a discrete integral-reflection representation of the double affine Hecke algebra of type $C^\vee C$ at the critical level q=1, to endow the open finite $q$-boson system with integrable boundary interactions at the lattice ends. It is shown that the Bethe Ansatz entails a complete basis of eigenfunctions for the commuting quantum integrals in terms of Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials.
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 $q$-boson system endowed with diagonal open-end boundary interactions.
Starting from a recently found branching formula for the six-parameter family of symmetric Macdonald-Koornwinder polynomials, we arrive by degeneration at corresponding branching rules for symmetric hypergeometric orthogonal polynomials of Wilson, continuous Hahn, Jacobi, Laguerre, and Hermite type.
We present an explicit difference equation for the Heckman-Opdam hypergeometric function associated with root systems. Via a confluent hypergeometric limit, an analogous difference equation is obtained for the class-one Whittaker function diagonalizing the open quantum Toda chain.
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 set-up.
We present an explicit branching formula for the six-parameter Macdonald-Koornwinder polynomials with hyperoctahedral symmetry.
We endow Ruijsenaars' open difference Toda chain with a one-sided boundary interaction of Askey-Wilson type and diagonalize the quantum Hamiltonian by means of deformed hyperoctahedral $q$-Whittaker functions that arise as a $t=0$ degeneration of the Macdonald-Koornwinder multivariate Askey-Wilson polynomials. This immediately entails the quantum integrability, the bispectral dual system, and the $n$-particle scattering operator for the chain in question.
For any admissible pair of irreducible reduced crystallographic root systems, we present discrete orthogonality relations for a finite-dimensional system of Macdonald polynomials with parameters on the unit circle subject to a truncation relation.
We introduce a representation of the double affine Hecke algebra at the critical level q=1 in terms of difference-reflection operators and use it to construct an explicit integrable discrete Laplacian on the Weyl alcove corresponding to an element in the center. The Laplacian in question is to be viewed as an integrable discretization of the conventional Laplace operator on Euclidian space perturbed by a delta-potential supported on the reflection hyperplanes of the affine Weyl group. The Bethe Ansatz method is employed to show that our discrete Laplacian and its commuting integrals are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions.
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.
We present a hierarchy of commuting operators in Fock space containing the q-boson Hamiltonian on $\mathbb{Z}$ and show that the operators in question are simultaneously diagonalized by Hall-Littlewood functions. As an application, the n-particle scattering operator is computed.
We introduce an explicit representation of the double affine Hecke algebra (of type $A_1$) at $q=1$ that gives rise to a periodic counterpart of a well-known Fourier transform associated with the affine Hecke algebra.