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On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

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Abstract

A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The location of the essential spectrum of H is described. The existence of infinitely many eigenvalues (resp. the finiteness of eigenvalues) below the bottom τess(H) of the essential spectrum of H is proved for the case where the associated Friedrichs model has a threshold energy resonance (resp. a threshold eigenvalue). For the number N(z) of eigenvalues of H lying below z < τess(H) the following asymptotics is found

$$\lim\limits_{z \to \tau_{\rm ess}(H)-0}\frac{N(z)}{|\log |z-\tau_{\rm ess}(H)||}={U}_0 (0<{U}_0 <\infty).$$

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115, Bonn, Germany

    Sergio Albeverio

  2. SFB 611, Bonn, BiBos, Bielefeld–Bonn, Germany

    Sergio Albeverio

  3. CERFIM, Locarno and Acc.Arch, USI, Switzerland

    Sergio Albeverio

  4. Samarkand University, University Boulevard 15, 703004, Samarkand, Uzbekistan

    Saidakhmat N. Lakaev & Tulkin H. Rasulov

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

    Saidakhmat N. Lakaev

Authors
  1. Sergio Albeverio
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  2. Saidakhmat N. Lakaev
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  3. Tulkin H. Rasulov
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Correspondence to Saidakhmat N. Lakaev.

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Subject Classification: Primary: 81Q10, Secondary: 35P20, 47N50.

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Albeverio, S., Lakaev, S.N. & Rasulov, T.H. On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics. J Stat Phys 127, 191–220 (2007). https://doi.org/10.1007/s10955-006-9240-6

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  • DOI: https://doi.org/10.1007/s10955-006-9240-6

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