\delta_{\sigma,\sigma'} \delta(E-E') \,\,\, .$$ In the case of two and three dimensional leads one can separate longitudinal and transverse particle motion. Since the transverse motion is quantized, the wave function relative to the plane perpendicular to the direction of transport is characterized by a set of quantum numbers which identifies the channels of the lead. Such channels are referred to as open when the corresponding longitudinal wave vectors are real, since they correspond to propagating modes. Note that the case of a single open channel corresponds to a one dimensional lead.

Let us now turn the attention to the probability distribution for the transfer of charges. Following Ref. [@muzykanskii94], within the scattering approach the characteristic function of the probability distribution for the transfer of particles in a structure attached to $n$ leads at a given energy $E$ can be written as $$\chi_E(\vec{\lambda_{\uparrow}},\vec{\lambda_{\downarrow}})=
\langle \prod_{j=1,n} e^{i\lambda_{j\uparrow}
\hat{N}_I^{j\uparrow}}~  e^{\lambda_{j\downarrow} \hat{N}_I^{j\downarrow}} ~
\prod_{j=1,n}
e^{-i\lambda_{j\uparrow} \hat{N}_O^{j\uparrow}}  e^{{-i\lambda_{j\downarrow}
\hat{N}_O^{j\downarrow}}} \rangle ~,
\label{chiE}$$ where the brackets $\langle ... \rangle$ stand for the quantum statistical average over the thermal distributions in the leads. Assuming a single channel per lead, $\hat{N}_{I(O)}^{j\sigma}$ is the number operator for incoming (outgoing) particles with spin $\sigma$ in lead $j$ and $\vec{\lambda_{\uparrow}}$, $\vec{\lambda_{\downarrow}}$ 