of ${{\cal I}}(g)$ (cf. [@rs1]): $$\begin{gathered}
\label{sval}
{{\cal I}}(g)=\sum_{n=0}^{\infty}\nu_n(g) e_n(g)\otimes e_n(g'),\qquad \nu_0\ge \nu_1\ge \cdots > 0,
\qquad \sum_{n=0}^{\infty}\nu_n^2<\infty,
\\
(e_m,e_n)={\delta}_{mn},\qquad m,n\in{{\mathbb N}}.\nonumber\end{gathered}$$ (All singular values are positive, since ${{\cal I}}(g)$ has trivial kernel and dense range for $g\in\Pi_r$, cf. (\[KerI\]). To appreciate the restriction on $s_g$, note that ${{\cal I}}(g)$ has rank 1 for $s_g=0$, cf. (\[cS\]).)

This result entails in particular that $H(g)$ and $H(g')$ have solely discrete spectrum, the two spectra being equal. To be sure, in the case at hand it follows from well-known Schrödinger operator theory that $H(g)$ can be reinterpreted as a self-adjoint operator that has an ONB of eigenvectors, provided $g$ is suitably restricted. The main novelty is that these eigenvectors are just those occurring for ${{\cal I}}(g)$, and that this fixes the self-adjoint extensions of the Schrödinger operator $H(g)$ in case it is not essentially self-adjoint on $C_0^{\infty}((0,\pi/2r))$ (namely, for $g_0$ and/or $g_1$ in $(-1/2,3/2)$). As a bonus, this yields a hidden spectral invariance under permutations of four new parameters, which implies spectral consequences for special cases that are quite startling. Last but not least, the state of affairs for the present case gives rise to a simple paradigm for the ‘relativistic’ (analytic 