{
  "id": "1506.06902",
  "version": "v1",
  "published": "2015-06-23T08:23:39.000Z",
  "updated": "2015-06-23T08:23:39.000Z",
  "title": "A bispectral q-hypergeometric basis for a class of quantum integrable models",
  "authors": [
    "Pascal Baseilhac",
    "Xavier Martin"
  ],
  "comment": "33 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.QA"
  ],
  "abstract": "For the class of quantum integrable models generated from the $q-$Onsager algebra, a basis of bispectral multivariable $q-$orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with $N$ variables and $N+3$ parameters introduced by Gasper and Rahman [GR1] generate a family of infinite dimensional irreducible modules for the $q-$Onsager algebra, whose fundamental generators are realized in terms of the multivariable $q-$difference and difference operators proposed by Iliev [Il]. Raising and lowering operators extending those of Sahi [Sa2] are also constructed. In a second part, finite dimensional irreducible modules are constructed and studied for a certain class of parameters and if the $N$ 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 $q-$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 $q=1$. This framework provides a $q-$hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions ($q\\neq 1$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-23T08:23:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum integrable models",
      "bispectral q-hypergeometric basis",
      "onsager algebra",
      "open xxz spin chain",
      "infinite dimensional irreducible modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606902B"
    }
  }
}