Constructing Hadamard States via an Extended Møller Operator

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 Møller 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 χ. 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 ξ is complete and fulfilling both LξV=0andLξχ=0. Eventually, we discuss a sufficient condition to remove on static spacetimes, the dependence on the cutoff via a suitable adiabatic limit.