  We consider real scalar field theories whose dynamics is ruled by normally
hyperbolic operators differing only by a smooth potential $V$. By means of an
extension of the standard definition of M{\o}ller operator, we construct an
isomorphism between the associated spaces of smooth solutions and between the
associated algebras of observables. On the one hand such isomorphism is
non-canonical since it depends on the choice of a smooth time-dependant cut-off
function. On the other hand, given any quasi-free Hadamard state for a theory
with a given $V$, such isomorphism allows for the construction of another
quasi-free Hadamard state for a different potential. The resulting state
preserves also the invariance under the action of any isometry, whose
associated Killing field commutes with the vector field built out of the normal
vectors to a family of Cauchy surfaces, foliating the underlying manifold.
Eventually we discuss a sufficient condition to remove on static spacetimes the
dependence on the cut-off via a suitable adiabatic limit.
