  The Heun equation can be rewritten as an eigenvalue equation for an ordinary
differential operator of the form $-d^2/dx^2+V(g;x)$, where the potential is an
elliptic function depending on a coupling vector $g\in{\mathbb R}^4$.
Alternatively, this operator arises from the $BC_1$ specialization of the
$BC_N$ elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev
system). Under suitable restrictions on the elliptic periods and on $g$, we
associate to this operator a self-adjoint operator $H(g)$ on the Hilbert space
${\mathcal H}=L^2([0,\omega_1],dx)$, where $2\omega_1$ is the real period of
$V(g;x)$. For this association and a further analysis of $H(g)$, a certain
Hilbert-Schmidt operator ${\mathcal I}(g)$ on ${\mathcal H}$ plays a critical
role. In particular, using the intimate relation of $H(g)$ and ${\mathcal
I}(g)$, we obtain a remarkable spectral invariance: In terms of a coupling
vector $c\in{\mathbb R}^4$ that depends linearly on $g$, the spectrum of
$H(g(c))$ is invariant under arbitrary permutations $\sigma(c)$, $\sigma\in
S_4$.
