  An important class of resonance problems involves the study of perturbations
of systems having embedded eigenvalues in their continuous spectrum. Problems
with this mathematical structure arise in the study of many physical systems,
e.g. the coupling of an atom or molecule to a photon-radiation field, and Auger
states of the helium atom, as well as in spectral geometry and number theory.
We present a dynamic (time-dependent) theory of such quantum resonances. The
key hypotheses are (i) a resonance condition which holds generically
(non-vanishing of the {\it Fermi golden rule}) and (ii) local decay estimates
for the unperturbed dynamics with initial data consisting of continuum modes
associated with an interval containing the embedded eigenvalue of the
unperturbed Hamiltonian. No assumption of dilation analyticity of the potential
is made. Our method explicitly demonstrates the flow of energy from the
resonant discrete mode to continuum modes due to their coupling. The approach
is also applicable to nonautonomous linear problems and to nonlinear problems.
We derive the time behavior of the resonant states for intermediate and long
times. Examples and applications are presented. Among them is a proof of the
instability of an embedded eigenvalue at a threshold energy under suitable
hypotheses.
