  In the perturbative treatment of interacting quantum field theories, if the
interaction Lagrangian changes adiabatically in a finite interval of time,
secular growths may appear in the truncated perturbative series also when the
interaction Lagrangian density is returned to be constant. If this happens, the
perturbative approach does not furnish reliable results in the evaluation of
scattering amplitudes or expectation values. In this paper we show that these
effects can be avoided for adiabatically switched-on interactions, if the
spatial support of the interaction is compact and if the background state is
suitably chosen. We start considering equilibrium background states and show
that, when thermalisation occurs (interaction Lagrangian of spatial compact
support), secular effects are avoided. Furthermore, no secular effects pop up
if the limit where the Lagrangian is supported everywhere in space is taken
after thermalisation (large time limit), in contrast to the reversed order.
This result is generalized showing that if the interaction Lagrangian is
spatially compact, secular growths are avoided for generic background states
which are only invariant under time translation and to states whose explicit
dependence of time is not too strong. Finally, as an example, we apply the
presented theorems to study a complex scalar and a Dirac field in a classical
external electromagnetic potential, on a background KMS state, to manifest that
a spatially compact supported interaction does not give rise to secular
growths.
