Resonances appearing by perturbation of embedded non-degenerate eigenvalues
are studied in the case when the Fermi Golden Rule constant vanishes. Under
appropriate smoothness properties for the resolvent of the unperturbed
Hamiltonian, it is proved that the first order Rayleigh-Schr\"odinger expansion
exists. The corresponding metastable states are constructed using this
truncated expansion. We show that their exponential decay law has both the
decay rate and the error term of order ε4, where ε is
the perturbation strength.Comment: To appear in Commun. Math. Phy