In  \[Numerical analysis\], we shall use these scattering matrices to build the scattering matrix $S$ associated to the *global* system made of $N$ quantum dots. For the present, let us assume that we are given the scattering matrix of the global system: $$S(E) = \left(S_{ij;mn}(E)\right)~,$$ where $i,j \in \{\L, \R, 1, \dots, N\}$ and for each couple $(i,j)$ the indices $m \in \{1, \dots, M_i\}$ and $n \in \{1, \dots, M_j\}$ number the channels in terminal $i$ and $j$, respectively. Therefore, the matrix element $S_{ij;mn}(E)$ is the probability *amplitude* that a particle with energy $E$ incident in channel $n$ in terminal $j$ is transmitted into channel $m$ in terminal $i$.

Being given $S(E)$, one can define the *total* transmission probability $t_{ij}(E)$ that a particle with energy $E$ goes from terminal $j$ to terminal $i$ [@Buttiker-Symmetry]: $$\label{transmission prob}
t_{ij}(E) = \sum_{m=1}^{M_i}
\sum_{n=1}^{M_j} |S_{ij;mn}(E)|^2~.$$

Throughout this paper, we always assume that the following holds.

For each $E > 0$, the complex matrix $S(E)$ is *unitary*.\[Assumption S\]

On the other hand, we do *not* assume the scattering matrix $S(E)$ to be symmetric. Using the unitarity of $S(E)$ one easily obtains the following results: $$\sum_{i} t_{ij}(E) = M_j~, \hspace{2mm} \forall j \hspace{7mm} \mbox{and} \hspace{7mm} 