I am playing with the spectral representation of a BCS gap function and I have trouble verifying causality properties. I find a divergence and I don't know what is the problem.

Assume the gap function is written $\Delta^{\text{BCS}}(\omega) = \Delta_1(\omega) + i\Delta_2(\omega)$ with $\Delta_1$ and $\Delta_2$ the real and imaginary parts, respectively. Thus, causality implies $$ \Delta^{\text{BCS}}(\omega) = \int_{-\infty}^{\infty}\frac{d\omega'}{\pi} \frac{\Delta_2(\omega')}{\omega'-\omega}, \ \Delta_1(\omega) = P \int_{-\infty}^{\infty} \frac{d\omega'}{\pi} \frac{\Delta_2(\omega')}{\omega'-\omega}, \ \Delta_2(\omega) = P\int_{-\infty}^{\infty} \frac{d\omega'}{\pi} \frac{\Delta_1(\omega')}{\omega'-\omega}, $$ where $P$ is the principal part. I think this is pretty standard, but I'm not so sure about the principal part. In this paper, they omit it.

For the BCS gap function, I assume a frequency-dependent gap function of the form $$\Delta_1(\omega) = \left\{ \begin{array}{cc} \Delta_0 & \text{for} \ \omega \in ]-\omega_c, \omega_c[ \\ 0 & \text{otherwise,} \end{array}\right.$$ so a constant gap function in frequencies, with a cutoff of $\omega_c$.

From the causality equation, I find $$ \Delta_2(\omega) = - \frac{\Delta_0}{\pi} \ln \Big \lvert \frac{\omega-\omega_c}{\omega+\omega_c} \Big \rvert. $$

I calculated the principal part like $$ \pi \Delta_2(\omega) = \lim_{\epsilon,\epsilon'\rightarrow0^+} \left[ \int_{-\infty}^{-\omega_c-\epsilon} \frac{d\omega'}{\omega'-\omega} \ln\left[ \frac{\omega'-\omega_c}{\omega'+\omega_c} \right] + \int_{-\omega_c+\epsilon}^{\omega-\epsilon'} \frac{d\omega'}{\omega'-\omega} \ln \left[ \frac{\omega_c - \omega'}{\omega_c+\omega'}\right] + \int_{\omega+\epsilon'}^{\omega_c-\epsilon} \frac{d\omega'}{\omega'-\omega} \ln \left[ \frac{\omega_c - \omega'}{\omega_c+\omega'} \right] + \int_{\omega_c+\epsilon}^{\infty} \frac{d\omega'}{\omega'-\omega} \ln \left[ \frac{\omega'-\omega_c}{\omega'+\omega_c} \right] \right]. $$

Now, if I try to recalculate $\Delta_1$ from the right causality equation and from $\Delta_2$, it does not quite work. First, I found from WolframAlpha that I can use $$\int_c^d \frac{d\omega'}{\omega'-\omega} \ln \left[\frac{\omega' - \omega_c}{\omega' + \omega_c} \right] = \ln\left[ \frac{d-\omega}{c-\omega} \right] \ln\left[ \frac{\omega-\omega_c}{\omega+\omega_c} \right] - \sum_{k=1}^{\infty} \frac{1}{k^2}\left( \frac{1}{(\omega-\omega_c)^k}-\frac{1}{(\omega+\omega_c)^k} \right) \left( (\omega-d)^k - (\omega-c)^k\right). $$ It seems to be true for any domain, but I somehow doubt it. Now to calculate the integral, I apply the principal part like above, use the last expression to calculate each parts and if I assume $\omega \in ] -\omega_c, \omega_c[$, I find $$ \Delta_1(\omega) = -\frac{\Delta_0}{\pi^2}\ln \left[ \frac{\omega_c+\omega}{\omega_c-\omega}\right] \ln[-1] + \lim_{b\rightarrow\infty} \sum_{k\in 2\mathbb{Z}+1} \frac{2b^k}{k^2} \left( \frac{1}{(\omega-\omega_c)^k}- \frac{1}{(\omega+\omega_c)^k} \right), $$ which is clearly divergent in the second term. Moreover, since $\ln[-1] = (2n+1)\pi i, \ n \in \mathbb{N}$, the first term looks like the imaginary part although there is an arbitrary $n$ and also I did take the principal part so it shouldn't be there (otherwise it can look like the general equation, the first above).

I've been trying this to much and don't really see any weak points I can work on, which is why I'm kindly asking your help. Perhaps I should post this on the Mathematical Stack Exchange? Although I could find anything, I does feel like this should be covered in some superconductivity courses. Thank you!


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