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The Threshold Effects for the Two-Particle Hamiltonians on Lattices

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Abstract

For a wide class of two-body energy operators h(k) on the d-dimensional lattice ℤ d, d≥3, k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k≠0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Germany

    S. Albeverio

  2. SFB 611, Bonn, BiBoS, Bielefeld - Bonn, Germany

    S. Albeverio

  3. CERFIM, Locarno and USI, Switzerland

    S. Albeverio

  4. Samarkand Division of Academy of Sciences of Uzbekistan,  , Uzbekistan

    S. N. Lakaev

  5. Department of Mathematics, University of Missouri, Columbia, MO, USA

    K.A. Makarov

  6. Samarkand State University, Samarkand, Uzbekistan

    S. N. Lakaev & Z.I. Muminov

Authors
  1. S. Albeverio
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  2. S. N. Lakaev
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  3. K.A. Makarov
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  4. Z.I. Muminov
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Correspondence to S. Albeverio.

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Communicated by B. Simon

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Albeverio, S., Lakaev, S., Makarov, K. et al. The Threshold Effects for the Two-Particle Hamiltonians on Lattices. Commun. Math. Phys. 262, 91–115 (2006). https://doi.org/10.1007/s00220-005-1454-y

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  • DOI: https://doi.org/10.1007/s00220-005-1454-y

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