{
  "id": "2412.11157",
  "version": "v1",
  "published": "2024-12-15T11:30:05.000Z",
  "updated": "2024-12-15T11:30:05.000Z",
  "title": "Polynomial potentials and nilpotent groups",
  "authors": [
    "W. Schweiger",
    "W. H. Klink"
  ],
  "comment": "33 pages, 8 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schr\\\"odinger operators of the form $H_N=X_0^2+V_N$, where $V_N=X_N^2+\\alpha X_{N-1}$ is a polynomial potential of degree $(2N-2)$ and $X_i$ are the generators of an irreducible representation of a particular nilpotent group $\\mathcal{G}_N$. Algebraization of the eigenvalue problem is achieved for eigenfunctions of the form $\\sum_{k=0}^M a_k X_2^k \\exp(-\\int dx\\, X_N)$. It is shown that the overdetermined linear system of equations for the coefficients $a_k$ has a nontrivial solution, if the parameter $\\alpha$ and $(N-3)$ Casimir invariants satisfy certain constraints. This general setting works for even $N\\geq 2$ and can also be applied to odd $N\\geq 3$, if the potential is symmetrized by considering it as function of $|x|$ rather than $x$. It provides a unified approach to quasi-exactly solvable polynomial interactions, including the harmonic oscillator, and extends corresponding results known from the literature. Explicit expressions for energy eigenvalues and eigenfunctions are given for the quasi-exactly solvable sextic, octic and decatic potentials. The case of $E=0$ solutions for general $N$ and $M$ is also discussed. As physical application, the movement of a charged particle in an electromagnetic field of pertinent polynomial form is shortly sketched.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-15T11:30:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial potential",
      "nilpotent group",
      "pertinent polynomial form",
      "casimir invariants satisfy",
      "energy eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}