Dirac-Delta from Normalization of Continuous Eigenfunctions

[MultipleSearching]

I'm following this paper by Kitaev on SL(2,R) representations and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy
$$\langle f_s | f_{s'} \rangle = \int_{0}^{1} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du. \tag{67}$$
The singular contribution of the integral arises at the endpoint $u=1$ of the integral, and in the limit $u \to 1$, the function $f_s(u)$ takes on the form
$$f_s(u) \approx a_s (1-u)^{1/2 + i s} + a_s^* (1-u)^{1/2 - i s}. \tag{70}$$
$f_{s'}(u)$ is also given by this expression but with $s \to s'$, where $s, s' \in \mathbb{R}^+$ and where $a_s, a_{s'} \in \mathbb{C}$ are complex coefficients in terms of $s, s'$, respectively, given by
$$a_s = \frac{\Gamma(-2i s)}{\Gamma\left(\frac{1}{2} - i s + l\right) \Gamma\left(\frac{1}{2} - i s + r\right)}.$$
Using these approximations, they immediately present in equation (70) the normalization condition
$$\langle f_s | f_{s'} \rangle = 4 \pi |a_s|^2 \delta(s - s'). \tag{70}$$
which is what I'm trying to obtain. My attempt at solving this involved splitting the integral into two, as follows
$$\int_{0}^{1} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du = \int_{0}^{1 - e^{-1/\varepsilon}} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) , du + \int_{1 - e^{-1/\varepsilon}}^{1} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du$$
and solving for the latter, which contains the divergent part. I used the $u \to 1$ approximation presented above and simplified the expression to a sum of 4 terms. Each term is an integral of the form (singular endpoint – problem arises here)
$$\mathcal{I} = \int_{1 - e^{-1/\varepsilon}}^{1} (1-u)^{i(s + s') - 1} , du = -\frac{i e^{-i(s + s')/\varepsilon}}{s + s'}.$$
After integrating all terms and using the following identities (in the distributional sense)
$$\lim_{a \to \infty} \frac{\sin(a x)}{\pi x} = \delta(x), \quad \lim_{a \to \infty} \cos(a x) = 0,$$
I got
$$\mathcal{I} = -4 \pi |a_s|^2 \delta(s - s').$$
I believe the problem comes from the way I solved the $\mathcal{I}$ integral, which leads to a sign discrepancy between what I obtained and the paper.

I've also tried the integral
$$\int_{0}^{1 - e^{-1/\varepsilon}} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du$$
Eventhough we're using an asymptotic expansion for $u \to 1$, I considered these limits since the dominant contribution to the Dirac delta function lies near $u=1$. This produces the correct sign for the normalization as seen in Kitaev's paper, but it also yields non-Dirac delta terms that don't cancel out. Namely
$$\int_0^1 \frac{2 du}{(1-u)^2} f_s(u) f_{s'}^*(u) = 4 \pi |a_s|^2 \delta(s-s') + \frac{4 \Im(a_s a_{s'})}{s+s'} + \frac{4 \Im(a_s a_{s'}^*)}{s-s'}$$

 

[jhonconnor]

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