Gibbs States of Interacting Systems of Quantum Anharmonic Oscillators

Abstract

A complete description of the equilibrium thermodynamic properties of an infinite system of interacting ν-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set \(\mathbb{L}\subset \mathbb{R}^d\), possibly irregular; the anharmonic potentials vary from site to site. The description is based on the representation of the Gibbs states in terms of path measures. In particular, it is stated that (a) the set of Gibbs measures \(\mathcal{G}^{\rm t}\) is non-void and compact; (b) every \(\mu \in \mathcal{G}^{\rm t}\) obeys exponential integrability estimates, the same for the whole \(\mathcal{G}^{\rm t}\); (c) every \(\mu \in \mathcal{G}^{\rm t}\) has a Lebowitz–Presutti type support; (d) \(|\mathcal{G}^{\rm t}| =1\) at high temperatures. In the case of ν = 1 and attractive interaction, the existence of phase transitions and uniqueness of Gibbs measures due to quantum effects are also described. Finally, it is shown that \(|\mathcal{G}^{\rm t}|=1\) at a non-zero external field.