Eqs. (3)–(7): normalizable modes of global $\mathrm{AdS}_{d+1}$ occur exactly at $|\omega|=2h_++2n+l$

Let $d\ge2$, $R>0$, $m\in\mathbb R$, $l\in\mathbb N$ and $\omega\in\mathbb R$, and let $\nu=\tfrac12\sqrt{d^2+4m^2R^2}$, $2h_+=\tfrac d2+\nu$. A radial profile $\chi$ is a normalizable mode of the free Klein–Gordon equation on global $\mathrm{AdS}_{d+1}$ with angular momentum $l$ and frequency $\omega$ (a nonzero solution of the radial equation that is regular at $r=0$ and has finite Klein–Gordon norm) if and only if there is $n\in\mathbb N$ with

and a constant $c\neq0$ such that $\chi=c\,\chi_{nl}$ on $(0,\pi/2)$, where $\chi_{nl}(r)=\sin^l r\,(\cos r)^{2h_+}P_n^{(l+d/2-1,\nu)}(\cos 2r)$.

This is the statement of eqs. (5)–(7) that free solutions exist at every frequency but are normalizable only on the discrete spectrum $\omega_{nl}$, and that the normalizable ones are the modes $\chi_{nl}$.

Formalization Note "Regular at the origin" means $|\chi|$ is bounded on $(0,\pi/4]$. "Finite Klein–Gordon norm" means $\tan^{d-1}r\,\chi(r)^2$ is integrable on $(0,\pi/2)$. The hypothesis $d\ge2$ gives $\nu\ge1$, so the $(\cos r)^{2h_-}$ branch is never normalizable.