  We analyze the transport properties of a double quantum dot device with both
dots coupled to perfect conducting leads and to a finite chain of N
non-interacting sites connecting both of them. The inter-dot chain strongly
influences the transport across the system and the Local Density of States of
the dots. We study the case of small number of sites, so that Kondo box effects
are present, varying the coupling between the dots and the chain. For odd N and
small coupling between the inter-dot chain and the dots, a state with two
coexisting Kondo regimes develops: the bulk Kondo due to the quantum dots
connected to leads and the one produced by the screening of the quantum dots
spins by the spin in the finite chain at the Fermi level. As the coupling to
the inter-dot chain increases, there is a crossover to a molecular Kondo
effect, due to the screening of the molecule (formed by the finite chain and
the quantum dots) spin by the leads. For even N the two-Kondo temperatures
regime does not develop and the physics is dominated by the usual competition
between Kondo and antiferromagnetism between the quantum dots. We finally study
how the transport properties are affected as N is increased. For the study we
used exact multi-configurational Lanczos calculations and finite U slave-boson
mean-field theory at T = 0. The results obtained with both methods describe
qualitatively and also quantitatively the same physics.
