We study the adiabatic time evolution of quantum resonances over time scales
which are small compared to the lifetime of the resonances. We consider three
typical examples of resonances: The first one is that of shape resonances
corresponding, for example, to the state of a quantum-mechanical particle in a
potential well whose shape changes over time scales small compared to the
escape time of the particle from the well. Our approach to studying the
adiabatic evolution of shape resonances is based on a precise form of the
time-energy uncertainty relation and the usual adiabatic theorem in quantum
mechanics. The second example concerns resonances that appear as isolated
complex eigenvalues of spectrally deformed Hamiltonians, such as those
encountered in the N-body Stark effect. Our approach to study such resonances
is based on the Balslev-Combes theory of dilatation-analytic Hamiltonians and
an adiabatic theorem for nonnormal generators of time evolution. Our third
example concerns resonances arising from eigenvalues embedded in the continuous
spectrum when a perturbation is turned on, such as those encountered when a
small system is coupled to an infinitely extended, dispersive medium. Our
approach to this class of examples is based on an extension of adiabatic
theorems without a spectral gap condition. We finally comment on resonance
crossings, which can be studied using the last approach.Comment: 35 pages. One remark added in section 3, and references updated. To
appear in Commun. Math. Phy
