I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.


For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev and Volkov et al. introduced the non-compact quantum dilogarithm
$$\DeclareMathOperator{\Dmi}{d\!}
e\_b(z)=\exp\left(\frac{1}{4}\int\_{C}\frac{e^{-2iz\zeta}}{\sinh(\zeta b)\sinh(\zeta b^{-1})\zeta}\Dmi\zeta\right),\quad|\operatorname{Im}(z)|<|\operatorname{Im}(c\_b)|$$
where $C$ is a contour going along the real line from $−\infty$ to $+\infty$ surpassing the origin in a small semi-circle from above and
$$c\_b=\frac{i(b+b^{-1})}{2}.$$
One can analytically continue $e\_b(z)$ to the whole complex plane by the following pair of functional equations
$$e\_b\left(z-\frac{ib^\pm}{2}\right)=\left(1+e^{2\pi zb^\pm}\right)e\_b\left(z+\frac{ib^\pm}{2}\right).$$
We can show that $e\_b$ is meromorphic with poles
$$\{c\_b+mib+nib^{-1}:m,n\in\mathbb{Z}\_{\geq0}\}.$$


Now given a separable Hilbert space $\mathcal{H}$ (you may take $\mathcal{H}$ to be $L^2(\mathbb{R})$ or $\ell^2(\mathbb{Z})$), with a pair of densely defined self-adjoint operators $\hat{P},\hat{X}$ satisfying the commutation relation $[\hat{P},\hat{X}]=\frac{1}{2\pi i}$. While restricted to the real line, we can see that $e\_b\in L^\infty(\mathbb{R})\cap C(\mathbb{R})$. So by Borel functional calculus, we have well-defined bounded operators $e\_b(\hat{P}),e\_b(\hat{X})$ and $e\_b(\hat{P}+\hat{X})$.


The pentagon identity states that
$$e\_b(\hat{P})e\_b(\hat{X})=e\_b(\hat{X})e\_b(\hat{P}+\hat{X})e\_b(\hat{P}).$$


However, the above statement is not satisfactory in the sense of functional analysis. What's the meaning of $[\hat{P},\hat{X}]$? Generally, we are not allowed to composite unbounded operators.


Moreover, the proof given in the appendix of <https://link.springer.com/article/10.1007/s002200100412> uses many singular integrations. For example, they use the "identity"
$$\phi\_+(x)\phi\_+(y)e^{2\pi ixy}=\int\_\mathbb{R}\phi\_+(z)\phi\_+(x-z)\phi\_+(y-z)e^{i\pi z^2}\Dmi z,$$
where $\phi\_+(z)$ is the inverse Fourier transform of $e\_b$. However, this inverse Fourier transform can only be taken in the sense of tempered distributions, and in fact $\phi\_+$ is not a function. Even if you use some functions to approximate $\phi\_+$ and deform the integral contour around singularities, the integration in RHS is still by no means Lebesgue integrable.


I'm asking for a mathematically rigorous proof. This may be well known to experts, but I never find a reference.


Thank you,


Estwald



Rigorous proof of the pentagon identity
[The pentagon relation for the quantum dilogarithm and quantized $M\_{0,5}$](https://arxiv.org/abs/0706.4054) by A.B. Goncharov explicitly attempts to be more rigorous than the paper cited in the OP.


