Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 104, 16 pages      arXiv:0805.0166      https://doi.org/10.3842/SIGMA.2009.104
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations

Ryu Sasaki a, Wen-Li Yang b, c and Yao-Zhong Zhang c
a) Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
b) Institute of Modern Physics, Northwest University, Xian 710069, P.R. China
c) School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

Received September 20, 2009, in final form November 10, 2009; Published online November 18, 2009

Abstract
Bethe ansatz formulation is presented for several explicit examples of quasi exactly solvable difference equations of one degree of freedom which are introduced recently by one of the present authors. These equations are deformation of the well-known exactly solvable difference equations of the Meixner-Pollaczek, continuous Hahn, continuous dual Hahn, Wilson and Askey-Wilson polynomials. Up to an overall factor of the so-called pseudo ground state wavefunction, the eigenfunctions within the exactly solvable subspace are given by polynomials whose roots are solutions of the associated Bethe ansatz equations. The corresponding eigenvalues are expressed in terms of these roots.

Key words: Bethe ansatz solution; quasi-exactly solvable models.

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