  We consider single-channel transmission through a double quantum dot system
that consists of two single dots coupled by a wire of finite length L. In order
to explain the numerically obtained results for a realistic double dot system
we explore a simple model. It consists, as the realistic system, of two dots
connected by a wire of length L. However, each of the two single dots is
characterized by a few energy levels only, and the wire is assumed to have only
one level whose energy depends on L. The transmission is described by using
S-matrix theory. The model explains in particular the splitting of the resonant
transmission peaks and the origin of the transmission zeros. The latter are
independent of the length of the wire. When the transmission zeros of the
single dots are of first order and both single dots are identical, those of the
double dot are of second order. First-order transmission zeros cause phase
jumps of the transmission amplitude by $\pi$, while there are no phase jumps
related to second-order transmission zeros. In this latter case, a phase jump
appears due to a resonance state whose decay width vanishes when crossing the
energy of the transmission zero.
