  Recent interesting experiments used scanning tunneling microscopy to study
systems involving Kondo impurities in quantum corrals assembled on Cu or noble
metal surfaces. The solution of the two-dimensional one-particle Schrodinger
equation in a hard wall corral without impurity is useful to predict the
conditions under which the Kondo effect can be projected to a remote location
(the quantum mirage). To model a soft circular corral, we solve this equation
under the potential W*delta(r-r0), where r is the distance to the center of the
corral and r0 its radius. We expand the Green's function of electron surface
states Gs0 for r<r0 as a discrete sum of contributions from single poles at
energies epsilon_i-I*delta_i. The imaginary part delta_i is the half-width of
the resonance produced by the soft confining potential, and turns out to be a
simple increasing function of epsilon_i. In presence of an impurity, we solve
the Anderson model at arbitrary temperatures using the resulting expression for
Gs0 and perturbation theory up to second order in the Coulomb repulsion U. We
calculate the resulting change in the differential conductance Delta dI/dV as a
function of voltage and space, in circular and elliptical corrals, for
different conditions, including those corresponding to recent experiments. The
main features are reproduced. The role of the direct hybridization between
impurity and bulk, the confinement potential, the size of the corral and
temperature on the intensity of the mirage are analyzed. We also calculate
spin-spin correlation functions.
