The Efimov effect of three-particle Schrödinger operators on a lattice. Asymptotics for the number of eigenvalues

Saidakhmat Lakaev Samarkand State University

The Efimov effect is one of the most remarkable results in the spectral analysis for three-particle Schrödinger operators on Euclidean space. Roughly speaking,the effect will be explained as follows:

If none of the three two-particle subsystem has bound states with negative energy, but at least two of them have a resonance state with zero energy, then the threeparticle system has an infinite number of three-particle bound states with negative energy, accumulating at zero.

This effect first discovered by Efimov in [1]. The mathematically rigorous proof of the result has been presented(see for instance [2-3]).

We consider a system of three-particle,which move in three-dimensional lattice Z^{3} and interact through a pair short-range negative potentials. The Hamiltonian of this system has form

H = △_{i} − ∑_{i<j} V_{ij},

where △_{i} is the lattice Laplacian and V_{ij} in the coordinate representation is the multiplication operator by function V_{ij}(r_{i} − r_{j}), V_{ij} ≥ 0. We reduce the investigation of the spectral properties of the operator H to the analysis of the family of self-adjoint,bounded operators (the three-particle discrete Schrödinger operators ) {H(K), K ∈ T^{3}}, acting in the Hilbert space of square-integrable functions defined on the torus, applying the Fourier transform and the decomposition into the direct operator integrals.

We assume that for the two-particle discrete Schrödinger operators h_{α}(k), k ∈ T^{3}, α = 1, 2, 3 some natural assumptions are fulfilled and establish the following results:

Let N_{0}(z) be the number of eigenvalues of the three-particle Schrödinger operator H(0) lying on the left from point z < 0.

(i)The N_{0}(z) obeys asymptotics

lim_{z→0} \frac{N_{0}(z)}{| log |z||} = C_{0} > 0,

where the C_{0} > 0 depend only on the masses of the particles.

Let U_{0} ⊂ T^{3} be sufficiently small punctured neighborhood of the point 0 ∈ T^{3}. (ii) For any K ∈ U_{0} the operator H(K) has only finite number eigenvalues lying

on the left-hand side of the essential spectrum.

We also shall point to some important spectral properties of the operators H(K), K ∈ T^{3},which are analogously or different for both of the Schrödinger operators on Euclidean space and lattice.

[1]V. Efimov: Nucl.Phys. A.210 (1973),157-158. [2]D. R. Yafaev: Math.USSR-Sb. 23 (1974),535-559. [3] A. V. Sobolev: Comm.Math. Phys. 156 (1993), 127–168.