  Let H be a self-adjoint operator bounded below by 1, and let V be a small
form perturbation such that RVS has finite norm, where R is the resolvent at
zero to the power 1/2 +epsilon, and S is the resolvent to the power
1/2-epsilon. Here, epsilon lies between 0 and 1/2. If the Gibbs state defined
by H is sufficiently regular, we show that the free energy is an analytic
function of V in the sense of Frechet, and that the family of density operators
defined in this way is an analytic manifold modelled on a Banach space.
