arXiv subjects
Publications and source records attributed to J. F. van Diejen.
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.
For any reduced crystallographic root system, we introduce a unitary representation of the (extended) affine Hecke algebra given by discrete difference-reflection operators acting in a Hilbert space of complex functions on the weight lattice. It is shown that the action of the center under this representation is diagonal on the basis of Macdonald spherical functions. As an application, we compute an explicit Pieri formula for these spherical functions.
We present explicit Pieri formulas for Macdonald's spherical functions (or generalized Hall-Littlewood polynomials associated with root systems) and their $q$-deformation the Macdonald polynomials. For the root systems of type $A$, our Pieri formulas recover the well-known Pieri formulas for the Hall-Littlewood and Macdonald symmetric functions due to Morris and Macdonald as special cases.
We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an explicit Pieri formula for the Macdonald polynomials in question. The simplest examples of our construction recover Macdonald's celebrated difference operators and associated Pieri formulas pertaining to the minuscule and quasi-minuscule weights. As further by-products, explicit expansions and Littlewood-Richardson type formulas are obtained for the Macdonald polynomials associated with a special class of small weights.
We introduce multivariate generalizations of the Bernstein-Szego polynomials, which are associated to the root systems of the complex simple Lie algebras. The multivariate polynomials in question generalize Macdonald's Hall-Littlewood polynomials associated with root systems. For the root system of type A1 (corresponding to the Lie algebra SL (2;C)) the classic Bernstein-Szego polynomials are recovered.
We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order.
We introduce an integrable lattice discretization of the quantum system of n bosonic particles on a ring interacting pairwise via repulsive delta potentials. The corresponding (finite-dimensional) spectral problem of the integrable lattice model is solved by means of the Bethe Ansatz method. The resulting eigenfunctions turn out to be given by specializations of the Hall-Littlewood polynomials. In the continuum limit the solution of the repulsive delta Bose gas due to Lieb and Liniger is recovered, including the orthogonality of the Bethe wave functions first proved by Dorlas (extending previous work of C.N. Yang and C.P. Yang).
We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.
To a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo Laplacians on the Weyl chamber. We develop the time-dependent scattering theory for these discrete pseudo Laplacians and determine the corresponding wave operators and scattering operators in closed form. As an application, we describe the scattering behavior of certain hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser models associated with the Macdonald polynomials.
We present a formula describing the asymptotics of a class of multivariate orthogonal polynomials with hyperoctahedral symmetry as the degree tends to infinity. The polynomials under consideration are characterized by a factorized weight function satisfying certain analyticity assumptions. As an application, the large-degree asymptotics of the Koornwinder-Macdonald BC_N-type multivariate Askey-Wilson polynomials is determined.
It is shown that the ground-state equilibrium configurations of the trigonometric BC-type Ruijsenaars-Schneider systems are given by the zeros of Askey-Wilson polynomials.
A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6ψ_6$ summation formula and its Dougall type ${}_5H_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts to an extension corresponding to the case of a nonreduced root system of certain summation identities associated to the reduced root systems that were recently conjectured by Aomoto and Ito and proved by Macdonald. By truncation, we obtain multidimensional analogues of the very-well-poised unilateral (basic) hypergeometric Rogers ${}_6ϕ_5$ and Dougall ${}_5F_4$ sums (both nonterminating and terminating). The terminating sums may be used to arrive at product formulas for the norms of recently introduced ($q$-)Racah polynomials in several variables.
Elementary properties of the Koornwinder-Macdonald multivariable Askey-Wilson polynomials are discussed. Studied are the orthogonality, the difference equations, the recurrence relations, and the orthonormalization constants for these polynomials. Essential in our approach are certain commuting difference operators simultaneously diagonalized by the polynomials.
Recent results are surveyed regarding the spectrum and eigenfunctions of the inverse square Calogero model with harmonic confinement and its relativistic analogue.
Multivariable generalizations of the continuous Hahn and Wilson polynomials are introduced as eigenfunctions of rational Ruijsenaars type difference systems with an external field.
Recent results are surveyed pertaining to the complete integrability of some novel n-particle models in dimension one. These models generalize the Calogero-Moser systems related to classical root systems. Quantization leads to difference operators instead of differential operators.
We present explicit generators of an algebra of commuting difference operators with trigonometric coefficients. The operators are simultaneously diagonalized by recently discovered q-polynomials (viz. Koornwinder's multivariable generalization of the Askey-Wilson polynomials). From the viewpoint of physics the algebra can be interpreted as consisting of the quantum integrals of a novel difference-type integrable sytem. This system generalizes the Calogero-Moser systems associated with non-exceptional root systems.