Heun operator of Lie type and the modified algebraic Bethe ansatz

Abstract

The generic Heun operator of Lie type is identified as a certain BC-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We also show that these Bethe roots are intimately associated with the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of O(3) and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.

Time and band limiting operator and Bethe ansatz

“…, L}. This type of relation between the spectra of the Heun and the time and band limiting operators has been discovered previously in different contexts [9,12]. We follow the approach used in [12], where the continuous time and band limiting operator is expressed as function of the confluent Heun operator, and consider the following equation,…”

“…Let us also mention that the separation of variable method has been developed in parallel to solve the same problem [35]. This connection between integrable systems and Heun operators opens a very interesting path and has already been used in the context of the computation of the entanglement entropy [9]. In this paper, we show that it allows to get insights into the time and band limiting problem by providing the Bethe ansatz for its associated Heun operator.…”

“…In many cases, including the original sinc kernel and its discrete counterpart, the commuting operator can be identified as an algebraic Heun operator [29,30]. It has been also used for the computation of the entanglement entropy of free Fermions on different chains [9,20,21] and on graphs associated to various association schemes [10,11,19]. These results provide an algebraic explanation of the existence of the tridiagonal matrix found by direct computations in [24].…”

The q-Racah polynomials from scalar products of Bethe states

“…Clearly, eigenbases for other examples of Leonard pairs and related orthogonal polynomials of the discrete Askey-scheme may be studied along the same line as limiting cases. For instance, a correspondence for the Racah type q = 1 may be studied starting from [BerCSV20,BerCCV22a]. Some perspectives are now presented.…”

The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz

“…Recently, the Bethe ansatz and its variants, which are cornerstones in the study of (quantum) integrable models, have found their way in representation theory. They have in particular appeared in two situations that bear a very close relation to the problem of finding the spectrum of an SO(3) scalar in U (su(3)): i. the diagonalization in [9] using the algebraic Bethe ansatz of the Lie-Heun operator introduced in [28] as labeling operator of a non-subgroup basis of O(3). ii.…”