content_image/368250.jpg content_image/368249.jpg content_image/368248.jpg content_image/368247.jpg content_image/368246.jpg
connected with a high frequency cut-off of the fluctuations. So all e-e processes we consider may be called Excitonically induced pairing. But all empirical information on high T$_{c}$ induced through e-e processes indicates that the ’excitons’ must be particle-hole fluctuations of the same one-particle excitations that pair up and such that they engender singularities in the single-particle spectra so that the distinction between particle-hole fluctuations of the fermions and the ’excitons’ is lost. 

## III. ELECTRON-PHONON INTERACTION PROMOTED T$_{c}$ 

## A. McMillan’s Expression for the Coupling constant 

For a general spectral function, 

<img src='content_image/368250.jpg'>

the Eliashberg equations require a numerical solution. For the limited purposes of this paper, it is enough to use the McMillan simplified solution [25], where by the transition temperature in the s-wave channel is given for the e-ph problem by 

with 

<img src='content_image/368249.jpg'>

<img src='content_image/368248.jpg'>

where N (0) is the density of states of un-renormalized electrons at the fermi-surface, < ω > is an average over the phonon frequencies, κ is a (suitably defined) average over the lattice stiffness, κ = M < ω 2 > , and < I 2 > is the scattering averaged over the spectral function of the phonons: − 1 − 1 

<img src='content_image/368247.jpg'>

where a spin-trace (in the singlet channel) of g ν $_{α,β}$( k , k + q ) g ν $_{γ,δ}$( − k , − k − q ) has been taken to obtain | g ν k , q | $^{2}$. 

It is important to note that the λ of Eq.(23) also gives the mass renormalization in the normal state due to phonons so that the renormalization of the specific heat coefficient due to interaction with phonons is 

<img src='content_image/368246.jpg'>

