3.8.3 SteadyState Kinetic Equations
Under steadystate condition the GREEN's functions depend on time
differences. One usually FOURIER transforms the time difference coordinate,
, to energy

(3.70) 
Under steadystate condition the
quantum kinetic equations, (3.64), (3.65), and
(3.69), can be
written as [60]:

(3.71) 

(3.72) 
where is the total selfenergy.
A similar transformation can be applied to selfenergies. However, to obtain
selfenergies one has to first apply LANGRETH's rules and then FOURIER
transform the time difference coordinate to energy.
We consider the selfenergies discussed in Section 3.6.
The evaluation of the HARTREE selfenergy due to electronelectron
interaction is straightforward, since it only includes the electron GREEN's
function. However, the lowestorder selfenergy due to electronphonon
interaction contains the products of the electron and phonon GREEN's
functions. Using LANGRETH's rules (Table 3.1) and then
FOURIER transforming the selfenergies
due to electronphonon interaction, (3.50) takes the form

(3.73) 
To calculate the retarded selfenergy, however,
it is more straightforward to FOURIER transform the relation
, see (3.52). By defining the broadening
function

(3.74) 
the retarded selfenergy is given by the convolution of
and
the FOURIER transform of the step function [33]

(3.75) 
where denotes the convolution. Therefore, the retarded selfenergy is
given by [116]

(3.76) 
where
stands for principal part.
M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors