On the Gibbs states for one-dimensional lattice Boson systems with a long-range interaction

We consider an infinite chain of interacting quantum (anharmonic) oscillators. The pair potential for the oscillators at lattice distance d is proportional to {d²[log(d+1)]F(d)}^-1 where ∑_r∈Z [rF(r)]^-1 < ∞. We prove that for any value of the inverse temperature β> 0 there exists a limiting Gibbs state which is translationally invariant and ergodic. Furthermore, it is analytic in a natural sense. This shows the absence of phase transitions in the systems under consideration for any value of the thermodynamic parameters.