  In the experiment of the quantum mirage, confinement of surface states in an
elliptical corral has been used to project the Kondo effect from one focus
where a magnetic impurity was placed, to the other empty focus. The signature
of the Kondo effect is seen as a Fano antiresonance in scanning tunneling
spectroscopy. This experiment combines the many-body physics of the Kondo
effect with the subtle effects of confinement. In this work we review the
essential physics of the quantum mirage experiment, and present new
calculations involving other geometries and more than one impurity in the
corral, which should be relevant for other experiments that are being made, and
to discern the relative importance of the hybridization of the impurity with
surface Vs and bulk Vb states. The intensity of the mirage imposes a lower
bound to Vs/Vb which we estimate. Our emphasis is on the main physical
ingredients of the phenomenon and the many-body aspects, like the dependence of
the observed differential conductance with geometry, which cannot be calculated
with alternative one-body theories. The system is described with an Anderson
impurity model solved using complementary approaches: perturbation theory in
the Coulomb repulsion U, slave bosons in mean field and exact diagonalization
plus embedding.
