Universality of the discrete spectrum asymptotics of the three-particle Schrödinger operator on a lattice

Mukhiddin I. Muminov – Faculty of Scince, Universiti Teknologi Malaysia (UTM) 81310 Skudai, Johor Bahru, Malaysia; mmuminov@mail.ru
Tulkin H. Rasulov – Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, Uzbekistan; rth@mail.ru

In the present paper, we consider the Hamiltonian H(K), K ∈ T3 := (−π; π]3 of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range potentials. We find a finite set Λ ⊂ T3 such that for all values of the total quasi-momentum K ∈ Λ the operator H(K) has infinitely many negative eigenvalues accumulating at zero. It is found that for every K ∈ Λ, the number N (K; z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim (z→−0) N (K; z)| log |z||-1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ.

Keywords: Three-particle Schrödinger operator, zero-range pair attractive potentials, Birman-Schwinger principle, the Efimov effect, discrete spectrum asymptotics.

PACS 02.30.Tb

DOI 10.17586/2220-8054-2015-6-2-280-293


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