{
  "id": "1511.09129",
  "version": "v1",
  "published": "2015-11-30T01:54:00.000Z",
  "updated": "2015-11-30T01:54:00.000Z",
  "title": "Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies",
  "authors": [
    "Gerardo Ariznabarreta",
    "Manuel Mañas"
  ],
  "comment": "38 pp",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP",
    "math.RT",
    "nlin.SI"
  ],
  "abstract": "Linear spectral transformations of orthogonal polynomials in the real line, and in particular Geronimus and Uvarov transformations, are extended to orthogonal polynomials depending on several real variables. Christoffel-Zhedanov type ormul{\\ae} for the perturbed orthogonal polynomials and their quasi-tau matrices are found for each perturbation of the original linear functional. These expressions are given in terms of quasi-determinants of bordered truncated block matrices and the 1D Christoffel-Zhedanov formul{\\ae}, in terms of quotient of determinants of combinations of the original orthogonal polynomials and their Cauchy transforms, are recovered. A new multispectral Toda hierarchy of nonlinear partial differential equations, which extend a previous one for which the multivariate orthogonal polynomials are reductions, is proposed. Wave and Baker functions, linear equations, Lax and Zakharov-Shabat equations, KP type equations, appropriate reductions, Darboux/linear spectral transformations, and bilinear equations involving linear spectral transformations are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-30T01:54:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J70",
      "15A23",
      "33C45",
      "37K10",
      "37L60",
      "42C05",
      "46L55"
    ],
    "keywords": [
      "multispectral toda hierarchy",
      "multivariate orthogonal polynomials",
      "nonlinear partial differential equations",
      "darboux/linear spectral transformations",
      "original orthogonal polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151109129A"
    }
  }
}