{
  "id": "1508.07747",
  "version": "v1",
  "published": "2015-08-31T10:03:17.000Z",
  "updated": "2015-08-31T10:03:17.000Z",
  "title": "Eigenfunction expansions for the Schrödinger equation with inverse-square potential",
  "authors": [
    "A. G. Smirnov"
  ],
  "comment": "20 pages, to appear in a special issue of Theoretical and Mathematical Physics dedicated to I.V. Tyutin on the occasion of his 75th birthday",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "The one-dimensional Schr\\\"odinger equation $-f\"+q_\\kappa f = Ef$ on the positive semi-axis with the potential $q_\\kappa(r)=(\\kappa^2-1/4)r^{-2}$ is considered. For each complex number $\\vartheta$, we construct a solution $u^\\kappa_\\vartheta(E)$ of this equation that is analytic in $\\kappa$ in a complex neighborhood of the interval $(-1,1)$ and, in particular, at the \"singular\" point $\\kappa = 0$. For $-1<\\kappa<1$ and real $\\vartheta$, the solutions $u^\\kappa_\\vartheta(E)$ determine a unitary eigenfunction expansion operator $U_{\\kappa,\\vartheta}\\colon L_2(0,\\infty)\\to L_2(\\mathbb R,\\mathcal V_{\\kappa,\\vartheta})$, where $\\mathcal V_{\\kappa,\\vartheta}$ is a positive measure on $\\mathbb R$. It is shown that every self-adjoint realization of the formal differential expression $-\\partial^2_r + q_\\kappa(r)$ for the Hamiltonian is diagonalized by the operator $U_{\\kappa,\\vartheta}$ for some $\\vartheta\\in\\mathbb R$. Using suitable singular Titchmarsh-Weyl $m$-functions, we explicitly find the measures $\\mathcal V_{\\kappa,\\vartheta}$ and prove their continuity in $\\kappa$ and $\\vartheta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-31T10:03:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schrödinger equation",
      "inverse-square potential",
      "unitary eigenfunction expansion operator",
      "formal differential expression",
      "complex number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150807747S"
    }
  }
}