\[section\] \[lem\][Theorem]{}

Introduction
============

In the first two papers of this series ([@HSI; @HSII], henceforth referred to as I and II), we initiated a detailed study of spectral properties of operators arising in the context of elliptic Calogero–Moser type systems. The operators at issue are not only the Hamiltonians defining these systems, but also families of Hilbert–Schmidt (HS) operators that seem unrelated at first sight. The relation consists in the kernels of the HS operators being eigenfunctions of differences of the Hamiltonians. As we have first detailed in our lecture notes [@EIS2] and also sketched in I, this relation can be exploited to derive novel information about the Hamiltonians, viewed as Hilbert space operators (as opposed to differential or difference operators).

The present paper is concerned with the Heun Hamiltonian $$\begin{gathered}
\label{Hg}
 H(g;x) \equiv -\frac{d^2}{dx^2}+V(g;x),  \qquad g=(g_0,g_1,g_2,g_3)\in{{\mathbb R}}^4,
\\
\label{Vg}
V(g;x)\equiv  \sum_{t=0}^3g_t(g_t-1)\wp(x+\omega_t;\pi/2r,i\alpha/2),\qquad \ r,\alpha>0,
 \end{gathered}$$ where $$\begin{gathered}
\omega_0\equiv 0,\qquad \omega_1\equiv \pi/2r,\qquad \omega_2\equiv i\alpha/2,\qquad \omega_3\equiv -\omega_1-\omega_2.\end{gathered}$$ (For a lucid account of elliptic and allied functions we recommend [@ww], whose conventions and notation we mostly follow.) It can also be viewed as the $BC_1$ ‘nonrelativistic’ elliptic Calogero–Moser Hamiltonian, the $BC_N$ case being the Inozemtsev system [@inoz]. There exists an extensive lore about 