attained for example for $\nu_j=\nu_{j+1}=\cdots =\nu_{j+k}$, by requiring that the associated $H(g)$-eigenvalues satisfy $\lambda_j<\lambda_{j+1}<\cdots <\lambda_{j+k}$ . With this ordering of the $e_n(g)$ understood from now on, there is a uniquely determined permutation $\tau_g$ of the nonnegative integers such that $$\begin{gathered}
\label{perm}
\lambda_{\tau_g(m)}(g)=E_m(g),\qquad m\in{{\mathbb N}},\end{gathered}$$ cf. (\[Hlam\]).

Next, we observe that we still have not completely fixed the choice of the basis vectors $e_n(g)$. Indeed, we have required real-valuedness of $e_n(g;x)$, but this leaves a sign undetermined. We may and will fix this ambiguity by requiring $$\begin{gathered}
d_n(g;0)>0,\qquad \forall \, n\in{{\mathbb N}},\end{gathered}$$ cf. (\[dnnz\]).

At this point the alert reader might (or rather should) object that the sign choice just detailed may be in conflict with the decomposition (\[Psirep\]). Indeed, once the ONB-functions $e_n(g;x)$ have been fixed, the ONB-functions $e_n(g';x)$ are uniquely determined, and the signs of the latter might differ from the above sign convention.

It follows from the last theorem of this section that this contingency does not arise. Its main result is, however, that $H(g)$ and $H(g')$ have the same spectrum.

\[theorem2.6\] The integral operator ${{\cal I}}(g)$ is real-analytic in $g$ on $\Pi_r$ in the Hilbert–Schmidt norm. With the above sign convention for $e_n(g;x)$ understood, it has singular 