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

Abstract

Abstract. We place the hyperbolic quantum Ruijsenaars-Schneider system with an exponential Morse term on a lattice and diagonalize the resulting nparticle 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.

The full phase space of a model in the Calogero–Ruijsenaars family

“…Similarly to its analogue in [7], the Hamiltonian (4.9) can be identified as an Inozemtsev type limit of a specialization of van Diejen's 5-coupling deformation of the hyperbolic BC n Sutherland Hamiltonian [14]. This fact suggests that it should be possible to extract the local form of dual Hamiltonians from [15] and references therein, which contain interesting results about closely related quantum mechanical systems and their bispectral properties. Indeed, in several examples, classical Hamiltonians enjoying action-angle duality correspond to bispectral pairs of Hamiltonian operators after quantization.…”

The action–angle dual of an integrable Hamiltonian system of Ruijsenaars–Schneider–van Diejen type

“…We finally remark that the quantum mechanical (bispectral) analogue of our dual pair should be understood. The recent paper by van Diejen and Emsiz [31] is certainly relevant for finding the answer to this question. We hope that our investigations will be developed in several directions in the future, including bispectral aspects withal.…”

On a Poisson–Lie deformation of the BC n Sutherland system

“…After we finished our work, there appeared a preprint [38] dealing with the quantum mechanics of a lattice version of a 4-parameter Inozemtsev type limit of van Diejen's trigonometric/hyperbolic system. The systems studied in [20] and in our paper correspond to further limits of specializations of this one.…”

“…The systems studied in [20] and in our paper correspond to further limits of specializations of this one. The statements about quantum mechanical dualities contained in [38] and its references should be related to classical dualities.…”