  We consider an "impurity" with a spin degree of freedom coupled to a finite
reservoir of non-interacting electrons, a system which may be realized by
either a true impurity in a metallic nano-particle or a small quantum dot
coupled to a large one. We show how the physics of such a spin impurity is
revealed in the many-body spectrum of the entire finite-size system; in
particular, the evolution of the spectrum with the strength of the
impurity-reservoir coupling reflects the fundamental many-body correlations
present. Explicit calculation in the strong and weak coupling limits shows that
the spectrum and its evolution are sensitive to the nature of the impurity and
the parity of electrons in the reservoir. The effect of the finite size
spectrum on two experimental observables is considered. First, we propose an
experimental setup in which the spectrum may be conveniently measured using
tunneling spectroscopy. A rate equation calculation of the differential
conductance suggests how the many-body spectral features may be observed.
Second, the finite-temperature magnetic susceptibility is presented, both the
impurity susceptibility and the local susceptibility. Extensive quantum
Monte-Carlo calculations show that the local susceptibility deviates from its
bulk scaling form. Nevertheless, for special assumptions about the reservoir --
the "clean Kondo box" model -- we demonstrate that finite-size scaling is
recovered. Explicit numerical evaluations of these scaling functions are given,
both for even and odd parity and for the canonical and grand-canonical
ensembles.
