Under steady-state condition the GREEN's functions depend on time
differences. One usually FOURIER transforms the time difference coordinate,
, to energy
3.8.3 Steady-State Kinetic Equations
Under steady-state condition the
quantum kinetic equations, (3.64), (3.65), and
(3.69), can be
written as :
where is the total self-energy.
A similar transformation can be applied to self-energies. However, to obtain
self-energies one has to first apply LANGRETH's rules and then FOURIER
transform the time difference coordinate to energy.
We consider the self-energies discussed in Section 3.6.
The evaluation of the HARTREE self-energy due to electron-electron
interaction is straightforward, since it only includes the electron GREEN's
function. However, the lowest-order self-energy due to electron-phonon
interaction contains the products of the electron and phonon GREEN's
functions. Using LANGRETH's rules (Table 3.1) and then
FOURIER transforming the self-energies
due to electron-phonon interaction, (3.50) takes the form
To calculate the retarded self-energy, however,
it is more straightforward to FOURIER transform the relation
, see (3.52). By defining the broadening
the retarded self-energy is given by the convolution of
the FOURIER transform of the step function 
where denotes the convolution. Therefore, the retarded self-energy is
given by 
stands for principal part.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors