We investigate multiple scattering of scalar waves by an ensemble of N
resonant point scatterers in three dimensions. For up to N=21 scatterers,
we numerically optimize the positions of the individual scatterers, such as to
maximize the total scattering cross section for an incoming plane wave, on the
one hand, and to minimize the decay rate associated to a long-lived scattering
resonance, on the other hand. In both cases, the optimimum is achieved by
configurations where all scatterers are placed on a line parallel to the
direction of the incoming plane wave. The associated maximal scattering cross
section increases quadratically with the number of scatterers for large N,
whereas the minimal decay rate -- which is realized by configurations that are
not the same as those that maximize the scattering cross section -- decreases
exponentially as a function of N. Finally, we also analyze the stability of
our optimized configurations with respect to small random displacements of the
scatterers. These results demonstrate that optimized configurations of
scatterers bear a considerable potential for applications such as quantum
memories or mirrors consisting of only a few atoms.Comment: 21 pages, 9 figure
