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

By Sergio Albeverio, S. Lakaev, Z. Muminov. 2003

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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.

Subfields: Applied Mathematics, Mathematical Physics

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