function $$\Psi_j(x|z)=\frac{1+2j}{\pi}\left[\left|\gamma(z)-x\right|^2e^{\phi(z)}+e^{-\phi(z)}\right]^{2j},\nonumber$$ where $(\phi,\gamma,\bar\gamma)$ are the Poincaré coordinates on H$_3^+$. The quantum operator lacks such a simple expression because of normal ordering. Nevertheless, in the large-$\phi$ regime the interaction vanishes and $\Phi_j(x|z)$ acquires the following form, $$\label{asymptotic}
\Phi_j(x|z)\sim:e^{-2(1+j)\phi(z)}:\delta\left(\gamma(z)-x\right)+B_j:e^{2j\phi(z)}:\left|\gamma(z)-x\right|^{4j},$$ with $$B_j=-\nu(b)^{1+2j}\frac{1+2j}{\pi}\frac{\Gamma(1+b^2(1+2j))}{\Gamma(1-b^2(1+2j))},\qquad\nu(b)=\pi\frac{\Gamma(1-b^2)}{\Gamma(1+b^2)}.\nonumber$$

As proved in [@Teschner:1997ft], the asymptotic expression given by Eq. (\[asymptotic\]) fixes a normalization of $\Phi_j(x|z)$ consistent with the following two-point function: $$\label{2pt}
\left\langle\Phi_{j_1}(x_1|z_1)\Phi_{j_2}(x_2|z_2)\right\rangle=\left[\delta\left(j_{12}^+\right)\delta\left(x_{12}\right)+B_{j_1}\delta\left(j_{12}\right)\left|x_{12}\right|^{4j_1}\right]\left|z_{12}\right|^{-4\Delta_{01}},$$ where $x_{12}=x_1-x_2$, $z_{12}=z_1-z_2$, $j_{12}=j_1-j_2$, $j_{12}^+=1+j_1+j_2$, and $\Delta_{01}$ stands for the conformal weight associated with $j_1$. The first term of this correlator is a contact term, while the one smeared over the boundary of H$_3^+$ is the so-called bulk or regular term.

As we have already pointed out, representations $\mathcal P_j$ and $\mathcal P_{-1-j}$, and thus $\mathcal R_j$ and $\mathcal R_{-1-j}$, are equivalent. The associated reflection symmetry is explicitly given by $$\label{reflexion}
\Phi_{j}(x|z)=R_j\left(\mathcal I_j\Phi_{-1-j}\right)(x|z),$$ where the reflection amplitude is $$\label{fifteen}
R_j=\frac{\pi}{1+2j}B_j=\nu(b)^{1+2j}\frac{\gamma(1+b^2(1+2j))}{b^2(1+2j)},$$ and the intertwining operator $\mathcal I_j$ is defined by $$\label{intertwining}
\left(\mathcal I_j\Phi_{-1-j}\right)(x|z)=\frac{1+2j}{\pi}\int_{\mathbb C}d^2x'\left|x-x'\right|^{4j}\Phi_{-1-j}(x'|z).$$ The overall factor in Eq. (\[intertwining\]) was chosen so that $\mathcal I_j\circ\mathcal I_{-1-j}=$Id, implying, by virtue of Eq. (\[stateoperator\]), its unitarity for $j\in -1/2+i\mathbb R$. Notice that a single reflection in the two-point function swaps the contact and the bulk terms, leaving the 