Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

Abstract

An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of 'symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.MSc: 81R50; 81R10; 81U15.

The rational Sklyanin algebra and the Wilson and para-Racah polynomials

“…It follows that for S to satisfy property (2.2), one must have that L decreases the degree of polynomials in λ x by one. Similarly, it follows from (2.4) that the case u i = δ i,1 will satisfy (2.2) if the case of u i = δ i,0 , corresponding to the S-Heun operator 1 2 (M 1 − M 2 ), is an operator that stabilizes the set of polynomials of a given degree.…”

“…These algebras have been the object of much attention from the perspective of algebraic geometry [28,34,18]. Classes of Heun operators can be defined [17] from the property that they increase by no more than one the degree of polynomials defined on certain continuous or discrete domains; they have been the focus of a continued research effort [29,2,33,10,30,3,5] with many applications [26,21,7,8,9,4,1]. A key observation for our purposes is that a special category of these operators, referred to as S-Heun operators, offers a path towards the identification of interesting Sklyanin-like algebras through the relations they realize.…”

The missing label of $\mathfrak{su}_3$ and its symmetry

“…As mentioned previously, there does not exist, in general, analytical formulas for the eigenvalues of X m and Y m . However, the connection with a Heun-Hahn operator suggests that it is possible to diagonalize X m with the help of the Bethe ansatz as it has been done in [3,5] for different algebraic Heun operators. The Heun-Hahn operator X m has been identified in the context of the two-body homogeneous rational Gaudin models in [8] and the nested Bethe ansatz developed in [18] for these models allows to obtain the eigenvalues of X m in terms of Bethe roots.…”

“…These translations are generated, for example, by two translations of order 6: say, t 1 sending m 1 to m 2 , and t 2 sending m 1 to m 2 + m ′ 2 − ℓ. These two translations satisfy t 3 1 = t 3 2 , so that the translation subgroup is of order 18. This checks with 144 = 8 × 18.…”

“…where n and ℓ are defined by (3) or (4). The 18 numbers are arranged as in the two magic squares in the Introduction.…”