{
  "id": "math-ph/0312026",
  "version": "v1",
  "published": "2003-12-10T15:23:10.000Z",
  "updated": "2003-12-10T15:23:10.000Z",
  "title": "Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics",
  "authors": [
    "Sergio Albeverio",
    "Saidakhmat N. Lakaev",
    "Zahriddin I. Muminov"
  ],
  "comment": "28 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice $\\Z^3$ and interacting via zero-range attractive potentials is considered. For the two-particle energy operator $h(k),$ with $k\\in \\T^3=(-\\pi,\\pi]^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of $h(k)$ for $k\\neq0$ is proven, provided that $h(0)$ has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schr\\\"{o}dinger operator $H(K), K\\in \\T^3$ being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number $N(0,z)$ of eigenvalues of H(0) lying below $z<0$ the following limit exists $$ \\lim_{z\\to 0-} \\frac {N(0,z)}{\\mid \\log\\mid z\\mid\\mid}=\\cU_0 $$ with $\\cU_0>0$. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum $K$ the finiteness of the number $ N(K,\\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the essential spectrum is established and the asymptotics for the number $N(K,0)$ of eigenvalues lying below zero is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-12-10T15:23:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "35P20",
      "47N50"
    ],
    "keywords": [
      "discrete spectrum asymptotics",
      "efimov effect",
      "schrödinger operators",
      "eigenvalue",
      "three-particle quasi-momentum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.ph..12026A"
    }
  }
}