A bispectral q-hypergeometric basis for a class of quantum integrable models
arXiv:1506.06902
Abstract
For the class of quantum integrable models generated from the Onsager algebra, a basis of bispectral multivariable orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with variables and parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the Onsager algebra, whose fundamental generators are realized in terms of the multivariable difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case . This framework provides a hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions ().
33 pages. v3: Theorem 3.3 removed. References reorganized according to the journal standards
References in corpus (32)
- Perfect state transfer in quantum spin networks
- Quantum symmetric Kac-Moody pairs
- Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms
- Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory
- Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields
- Functional relations from the Yang-Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions
- The half-infinite XXZ chain in Onsager's approach
- An integrable structure related with tridiagonal algebras
- A new (in)finite dimensional algebra for quantum integrable models
- Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
- Modified algebraic Bethe ansatz for XXZ chain on the segment - II - general cases
- Bispectrality of multivariable Racah-Wilson polynomials
- A new current algebra and the reflection equation
- Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields
- Exact solutions and elementary excitations in the XXZ spin chain with unparallel boundary fields
- Some algebra related to -and -polynomial association schemes
- Integrable Hierarchy of the Quantum Benjamin-Ono Equation
- Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
- Spin Chains with Non-Diagonal Boundaries and Trigonometric SOS Model with Reflecting End
- Correspondence between conformal field theory and Calogero-Sutherland model
- A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application
- Correlation functions of the half-infinite XXZ spin chain with a triangular boundary
- Koornwinder polynomials and the XXZ spin chain
- Racah Polynomials and Recoupling Schemes of
- An Exactly Solvable Spin Chain Related to Hahn Polynomials
- Generalized q-Onsager algebras and boundary affine Toda field theories
- Off-shell scalar products for the spin chain with open boundaries
- Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
- Integrable boundary interactions for Ruijsenaars' difference Toda chain
- Lax operator for Macdonald symmetric functions
- A classification of sharp tridiagonal pairs
- Mock Tridiagonal Systems