{
  "id": "1007.0737",
  "version": "v1",
  "published": "2010-07-05T18:08:21.000Z",
  "updated": "2010-07-05T18:08:21.000Z",
  "title": "The quantum $H_3$ integrable system",
  "authors": [
    "Marcos A. G. García",
    "Alexander V. Turbiner"
  ],
  "comment": "32 pages, 3 figures",
  "journal": "Intern.Journ.Mod.Phys. A 25 (2010) 5567-5594",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "The quantum $H_3$ integrable system is a 3D system with rational potential related to the non-crystallographic root system $H_3$. It is shown that the gauge-rotated $H_3$ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group $H_3$, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector $\\vec \\al\\ =\\ (1,2,3)$. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the $H_3$ Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of $H_3$-invariants \"polynomially\"-isospectral to the quantum $H_3$ model is defined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-05T18:08:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integrable system",
      "differential operators possessing finite-dimensional representations",
      "infinitely-many finite-dimensional invariant subspaces",
      "algebraic form"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1142/S0217732311034839",
      "journal": "Modern Physics Letters A",
      "year": 2011,
      "volume": 26,
      "number": 6,
      "pages": 433
    },
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011MPLA...26..433G"
    }
  }
}