J. F. van Diejen

Elliptic Selberg integrals

We introduce new Selberg-type multidimensional integrals built of Ruijsenaars elliptic gamma functions. We show that the vanishing of our integrals for a specific parameter hypersurface implies closed evaluation formulas valid for the full parameter space. The resulting integration formulas contain the Macdonald-Morris constant term identities for nonreduced root systems as special limiting cases.

Scattering from generalized point interactions using self‐adjoint extensions in Pontryagin spaces

Data transferring means separate and independent from the central processing unit of a computer system and operating in parallel with the central processing unit transfers data between a first memory area and a second memory area in a memory. The data transferring means comprises a first register for storing the address of the data of the first memory area from which the data is successively transferred to the second memory area. A second register stores the address of the data of a second memory area to which the data is successively transferred from the second memory area. A third register stores addresses of the group of data transferred from the first memory area to the second memory area. Transfer means transfers data directly to the second position in the second memory area designated by the address information of the second register. First arithmetic means connected to the first register and the second register modifies the address information of the first register and the second register by the information of the transfer data. Second arithmetic means connected to the third register modifies the address information of the transfer data.

Determinantal construction of orthogonal polynomials associated with root systems

We consider semisimple triangular operators acting in the symmetric component of the group algebra over the weight lattice of a root system. We present a determinantal formula for the eigenbasis of such triangular operators. This determinantal formula gives rise to an explicit construction of the Macdonald polynomials and of the Heckman-Opdam generalized Jacobi polynomials.

Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle

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).

Unit Circle Elliptic Beta Integrals

We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.

Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals

It is shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series. A conjectural evaluation formula for the integral then implies a closed summation formula for the series, generalizing both the multiple basic hypergeometric 8Φ7 sum of Milne-Gustafson type and the (one-dimensional) modular hypergeometric 8ε7 sum of Frenkel and Turaev. Independently, the modular invariance ensures the asymptotic correctness of our multiple modular hypergeometric summation formula for low orders in a modular parameter.

Diagonalization of the infinite q-boson system☆

We present a hierarchy of commuting operators in Fock space con- taining the q-boson Hamiltonian on Z and show that the operators in question are simultaneously diagonalized by Hall-Littlewood functions. As an applica- tion, the n-particle scattering operator is computed. The q-boson model constitutes a one-dimensional exactly solvable particle system in Fock space (BIK) based on the q-oscillator algebra (KS, Ch. 5). In the case of periodic boundary conditions (i.e. with particles hopping on the finite lattice Zm), the integrability, the spectrum, and the eigenfunctions of the Hamiltonian were analyzed by means of the algebraic Bethe Ansatz method (BIK). Remarkably, these eigenfunctions turn out to be Hall-Littlewood functions (T, K) (cf. also (J) for an alternative construction of Hall-Littlewood functions in Fock space based on deformed vertex operator algebras, with applications in the study of KP τ- functions arising from generating functions of weighted plane partitions (FW)). With the aid of explicit expressions for the commuting quantum integrals arising from an infinite-dimensional solution of the Yang-Baxter equation, it was very recently demonstrated (K) that the eigenvalue problem for the q-boson system on Zm is in fact equivalent to that of an integrable discretization (D) of the celebrated delta Bose gas on the circle (LL). The present work addresses the spectral problem and the integrability of the q-boson system on the infinite lattice Z. Specifically, we demonstrate that the eigenfunctions of this infinite q-boson system are again given by Hall-Littlewood functions and provide explicit formulas for a complete hierarchy of operators com- muting with the Hamiltonian; these formulas are natural infinite-dimensional ana- logues of the above-mentioned expressions in (K) for the finite q-boson system on Zm. Finally, the n-particle scattering operator is computed as an application of Ruijsenaars general scattering results in (R2). 2. The infinite q-boson system

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

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