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 show also that these Bethe roots are intimately associated to 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.

References in corpus (25)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Perfect Transfer of Arbitrary States in Quantum Spin Networks
• Mirror Inversion of Quantum States in Linear Registers
• Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions
• Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz
• Modified algebraic Bethe ansatz for XXZ chain on the segment - II - general cases
• Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof
• Properties of the entanglement Hamiltonian for finite free-fermion chains
• Free-Fermion entanglement and orthogonal polynomials
• Algebraic Heun operator and band-time limiting
• Algebraization of difference eigenvalue equations related to $U_q(s{l}_2)$
• Tridiagonalization and the Heun equation
• Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary
• The Heun-Askey-Wilson algebra and the Heun operator of Askey-Wilson type
• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries
• The Heun operator as a Hamiltonian
• Bethe Ansatz Equations of XXZ Model and q-Sturm-Liouville Problems
• FRT presentation of classical Askey-Wilson algebras
• FRT presentation of the Onsager algebras
• Heun algebras of Lie type
• Equivalent T-Q relations and exact results for the open TASEP
• The Heun-Racah and Heun-Bannai-Ito algebras
• A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
• Entanglement of Free Fermions on Hadamard Graphs

Cited by in corpus (8)

• Entanglement of inhomogeneous free fermions on hyperplane lattices
• Entanglement of Free Fermions on Johnson Graphs
• Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes
• Bethe ansatz diagonalization of the Heun-Racah operator
• The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz
• Computation of entanglement entropy in inhomogeneous free fermions chains by algebraic Bethe ansatz
• The missing label of ${\mathfrak{su}}_3$ and its symmetry
• Time and band limiting operator and Bethe ansatz