, Volume 5, Issue 4, pp 743-772

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

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Abstract.

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.

Communicated by Gian Michele Graf
Submitted 19/11/03, accepted 08/03/04