In the case of the polariton problem, we have seen that, even when neglecting Coulomb repulsion (as in the original BCS formulation), the potential $U(\omega )$ departs strongly from the Cooper potential and features two large attractive regions far from small energies, immediately followed by two strong repulsive windows. We extend the Bogoliubov method to a three-step approximation of this potential, such as displayed in Fig. 13, with, in reference to previous potentials, notations $\omega D$, $\omega C$ and $\omega B$ for the boundaries of the central, shallow attractive region, narrow, deep attractive region and repulsive region, respectively. This is a notation only and is not mean to be understood as referring to Debye, Coulomb, or Bogoliubov in any strict sense. Following the same premises, we approximate the gap equation by a three-step valued function $\Delta =(\Delta 1,\Delta 2,\Delta 3)T$. This approximation turns out to be an exceedingly good one in certain cases, such as the one displayed in Fig. 13, where the gap itself is also a three-steps function in good approximation. Here there is even more room to choose a parametrization of the $I$ matrix. We now give general guidelines on how to build this matrix. The simplest method is to fix $\xi $ on the lhs of Eq. (25) at the center of each region and, in the corresponding row, take for each column the potential that is sampled more by the difference $|\xi \u2212\xi \u2032|$. Refinements are possible, such as weighting elements of $I$ by coefficients which reflect how much time the variables $\xi $ and $\xi \u2032$ spend in the regions that determine the matrix equation. This problem has the following mathematical expression: how is the random variable $X\u2212Y$ distributed when $X$ (resp. $Y$) is uniformly distributed in an interval $[gi,gi+1]$ (resp. $[hi,hi+1]$). The solution is easily obtained as proportional (normalize to unity) to: Display Formula
$P(X\u2212Y=\theta )\u221d[max(gi,hi\u2212\theta )]\u2212min(gi+1,hi+1\u2212\theta )]2+[max(\theta +gi,hi)\u2212min(\theta +gi+1,hi+1]2.$(32)
This is easily obtained geometrically (the square root comes from Pythagoras’ theorem) and the problem results in finding the intersect of a line with the grid. There are two configurations. Working out the cases shows that Eq. (32) reduces to a triangular or a top-head truncated triangular distribution. The coefficients entering $I$ can then be taken as the potentials weighted by the area intersecting their corresponding region. The soundness of such an approach can be checked numerically to compare quantitatively various parametrizations. For simplicity, we shall here consider cases where only the dominant potential is considered. An example of such a gap equation $(I\u22121)\Delta =0$ is defined with: Display Formula$I=\u2212(V0I1V1I2\u2212VCI3V1I1V0I2V0I3\u2212VCI1V0I2V0I3),$(33)
where Display Formula$Ii=\u222b\u210f\omega i<\u210f\omega i>tanh[\xi /(2kBTC)]\xi 2+\Delta i2d\xi ,$(34)
and $Ii$ is integrated on the respective steps, as defined by the integral boundary conditions, i.e., $(\omega 1<,\omega 1>,\omega 2<,\omega 2>,\omega 3<,\omega 3>)=(0,\omega D,\omega D,\omega C,\omega C,\omega B)$.