Title | The number of eigenvalues of three-particle Schrödinger operators on lattices |

Publication Type | Journal Article |

Year of Publication | 2007 |

Authors | Albeverio, S, Dell'Antonio, G, Lakaev, SN |

Journal | J. Phys. A 40 (2007) 14819-14842 |

Abstract | We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and boson)on the three-dimensional lattice $\\\\Z^3$ and interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schr\\\\\\\"{o}dinger operator $H_{\\\\gamma}(K),$ $K$ being the total quasi-momentum and $\\\\gamma>0$ the ratio of the mass of fermion and boson.\\nWe choose for $\\\\gamma>0$ the interaction $v(\\\\gamma)$ in such a way the system consisting of one fermion and one boson has a zero energy resonance.\\nWe prove for any $\\\\gamma> 0$ the existence infinitely many eigenvalues of the operator $H_{\\\\gamma}(0).$ We establish for the number $N(0,\\\\gamma; z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$ \\\\lim_{z\\\\to 0-}\\\\frac{N(0,\\\\gamma;z)}{\\\\mid \\\\log \\\\mid z\\\\mid \\\\mid}={U} (\\\\gamma) .$$ Moreover, for all nonzero values of the quasi-momentum $K \\\\in T^3 $ we establish the finiteness of the number $ N(K,\\\\gamma;\\\\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,\\\\gamma;0)$ of eigenvalues below zero. |

URL | http://hdl.handle.net/1963/2576 |

DOI | 10.1088/1751-8113/40/49/015 |

Alternate Journal | Asymptotics for the number of eigenvalues of three-particle Schrödinger operators on lattices |

## The number of eigenvalues of three-particle Schrödinger operators on lattices

Research Group: