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Journal of Physics A Mathematical and General

The existence of many-particle bound states despite a pair interaction with positive scattering length

Bernhard Baumgartner

Published Nov 21, 1997• 13 citations
Publisher

Abstract

Examples of bound states are presented, for a system of three identical particles as well as for a system of particles on a lattice. The particles may be bosons as well as fermions. The interaction between them is given by a pair potential, which does not allow two-particle bound states, has positive scattering length and is not too different from realistic interatomic potentials.

DOI
10.1088/0305-4470/30/22/001

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Examples of bound states are presented, for a system of three identical particles as well as for a system of particles on a lattice. The particles may be bosons as well as fermions. The interaction between them is given by a pair potential, which does not allow two-particle bound states, has positive scattering length and is not too different from realistic interatomic potentials.
We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength μ1∈R\mu_1\in\mathbb{R}μ1​∈R , while the interaction between a fermion and one other particle is via an on-site potential with strength μ2∈R{\mu_2\in\mathbb{R}}μ2​∈R . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum K∈T1K\in\mathbb{T}^1K∈T1 . Furthermore, we show that both the bound state fμ1μ2(K; ⋅ , ⋅ )f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)fμ1​μ2​​(K;⋅,⋅) and its corresponding eigenvalue Eμ1μ2(K)E_{\mu_1\mu_2}(K)Eμ1​μ2​​(K) depend holomorphically on the quasimomentum.
We prove the existence of two-and three-particle bound states of the Schrödinger operators $h_μ(k),k\in \T^d$ and $H_μ(K),K\in \T^d$ associated to Hamiltonians hμ\mathrm{h}_μhμ​ and Hμ\mathrm{H}_μHμ​ of a system of two and three identical bosons on the lattice Zd,d=1,2\Z^d, d=1,2Zd,d=1,2 interacting via pairwise zero-range attractive μ<0μ<0μ<0 or repulsive μ>0μ>0μ>0 potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.
A method by Post, which has been used to calculate lower bounds on the energies of the lowest spatially symmetric or antisymmetric states of many-identical-particle systems, is shown to apply quite generally to the ground states of systems of bosons or fermions for any pair interactions whatever. In particular, the N -nucleon system is considered. No ordinary central pair potential was found to give the empirical binding energies of 2 H, 3 H and 4 He. For the exponential and square-well potentials the calculated bounds on the energies are closer than 1% over wide ranges of the potential parameters. The method justifies and generalizes certain `equivalent two-body' calculations originated by Wigner.