Lattice systems with a continuous symmetry - I. Perturbation theory for unbounded spins

We investigate a continuous Ising system on a lattice, equivalently an anharmonic crystal, with interactions: {Mathematical expression} We prove that the perturbation expansion for the free energy and for the correlation functions is asymptotic about λ=0, despite the fact that the reference system (λ=0) does not cluster exponentially. The results can be extended to more general systems of this type, e.g. an even polynomial semibounded from below instead of a quartic interaction. By a suitable scaling, λ corresponds to the temperature.