{
  "id": "2001.06128",
  "version": "v1",
  "published": "2020-01-17T01:02:05.000Z",
  "updated": "2020-01-17T01:02:05.000Z",
  "title": "Coupling constant dependence for the Schrödinger equation with an inverse-square potential",
  "authors": [
    "A. G. Smirnov"
  ],
  "comment": "48 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider the one-dimensional Schr\\\"odinger equation $-f''+q_\\alpha f = Ef$ on the positive half-axis with the potential $q_\\alpha(r)=(\\alpha-1/4)r^{-2}$. It is known that the value $\\alpha=0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\\partial^2_r + q_\\alpha(r)$ for the Hamiltonian have infinitely many eigenvalues for $\\alpha<0$ and at most one eigenvalue for $\\alpha\\geq 0$. For each complex number $\\vartheta$, we construct a solution $\\mathcal U^\\alpha_\\vartheta(E)$ of this equation that is entire analytic in $\\alpha$ and, in particular, is not singular at $\\alpha = 0$. For $\\alpha<1$ and real $\\vartheta$, the solutions $\\mathcal U^\\alpha_\\vartheta(E)$ determine a unitary eigenfunction expansion operator $U_{\\alpha,\\vartheta}\\colon L_2(0,\\infty)\\to L_2(\\mathbb R,\\mathcal V_{\\alpha,\\vartheta})$, where $\\mathcal V_{\\alpha,\\vartheta}$ is a positive measure on $\\mathbb R$. We show that each operator $U_{\\alpha,\\vartheta}$ diagonalizes a certain self-adjoint realization $h_{\\alpha,\\vartheta}$ of the expression $-\\partial^2_r + q_\\alpha(r)$ and, moreover, that every such realization is equal to $h_{\\alpha,\\vartheta}$ for some $\\vartheta\\in\\mathbb R$. Employing suitable singular Titchmarsh--Weyl $m$-functions, we explicitly find the spectral measures $\\mathcal V_{\\kappa,\\vartheta}$ and prove their smooth dependence on $\\alpha$ and $\\vartheta$. Using the formulas for the spectral measures, we analyse in detail how the transition through the point $\\alpha=0$ occurs for both the eigenvalues and the continuous spectrum of $h_{\\alpha,\\vartheta}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-17T01:02:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coupling constant dependence",
      "inverse-square potential",
      "schrödinger equation",
      "spectral measures",
      "unitary eigenfunction expansion operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}