On the number of eigenvalues of a model operator related to a system of three particles on lattices

Abstract

We consider a quantum mechanical system on a lattice in which three particles, two of them being identical, interact through a zero-range potential. We admit a very general form for the ‘kinetic’ part H0γ of the Hamiltonian, which contains a parameter γ to distinguish the two identical particles from the third one (in the continuum case this parameter would be the inverse of the mass). We prove that there is a value γ* of the parameter such that only for γ < γ* the Efimov effect (infinite number of bound states if the two-body interactions have a resonance) is absent for the sector of the Hilbert space which contains functions which are antisymmetric with respect to the two identical particles, while it is present for all values of γ on the symmetric sector. We comment briefly on the relation of this result with previous investigations on the Thomas effect. We also establish the following asymptotics for the number N(z) of eigenvalues z below Emin, the lower limit of the essential spectrum of H0. In the symmetric subspace whereas in the antisymmetric subspace where , are written explicitly as a function of the integral kernel of operators acting on ( is the unit sphere in