Metastable States When the Fermi Golden Rule Constant Vanishes

Resonances appearing by perturbation of embedded non-degenerate eigen-values 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-Schrodinger 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 epsilon(4), where epsilon is the perturbation strength.