operator $T(g)$, which (by the first assertion) is continuous in $g$ on $\Pi_r$ in the trace norm topology. As $\Pi_r$ is a connected set, it suffices to check positivity for one $g$ in $\Pi_r$. For $g=(0,0,1,1)$ we have $g'=g$ and ${{\cal I}}(g)$ is self-adjoint. Thus the numbers $\mu_m(g)$ are eigenvalues of ${{\cal I}}(g)$. These are easily determined explicitly, and they are all positive, cf. (\[nun\]).

For all of the special $g\in\Pi_r$ where we have calculated the numbers $\mu_m(g)$ explicitly (cf. Section \[section4\]), they are distinct and decreasing in $m$: $$\begin{gathered}
\mu_0(g)>\mu_1(g)>\mu_2(g)>\cdots >0.\end{gathered}$$ This entails in particular that the permutation $\tau_g$ is the identity permutation. It is an open problem whether these features hold true on all of $\Pi_r$.

A hidden $\boldsymbol{S_4}$ symmetry {#section3}
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In this section our purpose is to study, for a given coupling vector $g\in \Pi_r$, the couplings $\hat{g}$ for which the spectra of $H(g)$ and $H(\hat{g})$ are equal. To this end we note that we can establish equality of spectra by exploiting the core property of ${D(g_0,g_1)}$, cf. Lemma \[lemma2.3\]. First, since the action of the differential operator $H(g)$ on the domain ${D(g_0,g_1)}$ is invariant under taking $g_t\to 1-g_t$, and the domain does not depend on 