  We present a full introduction to the recent devised perturbation theory for
strong coupling in quantum mechanics. In order to put the theory in a proper
historical perspective, the approach devised in quantum field theory is rapidly
presented, showing how it implies a kind of duality in perturbation theory,
from the start. The approach of renormalization group in perturbation theory is
then presented. This method permits to resum secularities in perturbation
theory and makes fully algorithmical the resummation, transforming the
perturbation calculations in a step by step computational procedure. The
general theorem on which is founded a proper application of the strong coupling
expansion, based on a result in the quantum adiabatic theory, is then exposed.
This theorem gives the leading order of a strong coupling expansion. Then,
after the introduction of the principle of duality in perturbation theory that
puts in a proper context the quantum field theory method, the resulting theory
of the strong coupling expansion and the free picture are presented. An
algorithm for the computation of the perturbation series is finally given. This
approach has a lot of applications in fields as quantum optics, condensed
matter and so on, extending the original expectations of the quantum field
theory method. So, we give some examples of application for a class of
two-level systems that, in recent years, proved to be extremely important. One
of the most interesting concepts that can be obtained in this way is that of a
Quantum Amplifier (QAMP) that permits to obtain an amplification to the
classical level of the quantum fluctuations of the ground state of a single
radiation mode.
