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Explore the diagonalization of Heun-Askey-Wilson operator Leonard pairs using the algebraic Bethe ansatz in this 43-minute conference talk. Delve into the construction of eigenstates as Bethe states for various specializations and the generic case, examining the corresponding Bethe ansatz equations. Discover alternative presentations of Bethe equations using 'symmetrized' Bethe roots and learn how two families of on-shell Bethe states generate explicit bases for tridiagonal Leonard pair actions. Investigate the derivation of (in)homogeneous Baxter T-Q relations and explore realizations of the Heun-Askey-Wilson operator as second q-difference operators. Examine the connection between Q-polynomials and Askey-Wilson polynomials, leading to solutions for associated Bethe ansatz equations. Consider this analysis as a model for studying integrable systems generated from the Askey-Wilson algebra and its generalizations. Conclude by discussing the potential expression of qRacah polynomials as scalar products of Bethe states.
