{
  "id": "1511.07670",
  "version": "v1",
  "published": "2015-11-24T12:15:30.000Z",
  "updated": "2015-11-24T12:15:30.000Z",
  "title": "On the Number of Bound States of Finitely Many Point Interactions on Riemannian Manifolds",
  "authors": [
    "Fatih Erman"
  ],
  "comment": "23 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We first review the construction of the problem of a quantum particle interacting with $N$ attractive point $\\delta$-interactions in two and three dimensional Riemannian manifolds and improve it in a more rigorous way so that our somewhat heuristic arguments, which were used in our earlier works, are justified. We prove that the principal matrix is a matrix-valued holomorphic function on the region $\\mathcal{R}=\\{ z \\in \\mathbb{C}| \\Re(z) <0 \\}$ for two particular classes of Riemannian manifolds. We show that the essential spectrum of the Hamiltonian and that of the free Hamiltonian coincides. We finally give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-24T12:15:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bound states",
      "point interactions",
      "dimensional riemannian manifolds",
      "free hamiltonian coincides",
      "somewhat heuristic arguments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}