Self similar solutions of the evolution equation of a scalar field in an expanding geometry

We consider the functional Schrödinger equation for a self interacting scalar field in an expanding geometry. By performing a time dependent scale transformation on the argument of the field we derive a functional Schrödinger equation whose hamiltonian is time independent but involves a time-odd term associated to a constraint on the expansion current. We study the mean field approximation to this equation and generalize in this case for interacting fields, the solutions worked out by Bunch and Davies for free fields.