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

Abstract

Universality of the discrete spectrum asymptotics of the three-particle Schr¨ odinger operator on a lattice Mukhiddin I. Muminov1, Tulkin H. Rasulov2 1Faculty of Scince, Universiti Teknologi Malaysia (UTM) 81310 Skudai, Johor Bahru, Malaysia 2Faculty of Physics and Mathematics, Bukhara State University M. Ikbol str. 11, 200100 Bukhara, Uzbekistan mmuminov@mail.ru, rth@mail.ru PACS 02.30.Tb DOI 10.17586/2220-8054-2015-6-2-280-293 In the present paper, we consider the Hamiltonian H(K), K ?T3:= (-p;p]3of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range poten- tials. We find a finite set ??T3such that for all values of the total quasi-momentum K??the operator H(K)has infinitely many negative eigenvalues accumulating at zero. It is found that for every K??, the number N(K;z)of eigenvalues of H(K)lying on the left of z, z < 0,satisfies the asymptotic relation lim z?-0N(K;z)|log |z||-1=U0with 0<U0<8,independently on the cardinality of ?.