Annales Henri Poincaré

, Volume 5, Issue 4, pp 743–772

Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

Original paper

DOI: 10.1007/s00023-004-0181-9

Cite this article as:
Albeverio, S., Lakaev, S.N. & Muminov, Z.I. Ann. Henri Poincaré (2004) 5: 743. doi:10.1007/s00023-004-0181-9


The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice \( \mathbb{Z}^3 \) and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with \( k \in \mathbb{T}^3 = (-\pi, \pi]^3 \) the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k ≠ 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), \( k \in \mathbb{T}^3 \) 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

\( \lim_{z \to 0-}\, {N(0, z)\over |\log|z\|} = {\mathcal U}_0 \)

with \( {\mathcal U}_0 > 0. \) 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.

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Samarkand State UniversitySamarkandUzbekistan
  3. 3.Samarkand State UniversitySamarkandUzbekistan

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