difference operator) Heun case, where the usual self-adjoint extension theory is of no help and no previous spectral results are known, cf. the fourth paper in this series [@HSIV].

We proceed to sketch the organization and results of this paper in more detail. We begin Section \[section2\] by deriving features of functions in the range of ${{\cal I}}(g)$, assuming $g\in\Pi_r$. Then we show that ${{\cal I}}(g),g\in\Pi_r$, has dense range and trivial kernel. These crucial features readily follow from Lemma \[lemma2.1\], which asserts that the integral operator with kernel ${{\cal S}}(g;x,y)$ has kernel $\{ 0\}$ on $L^1([0,\pi/2r])$ whenever $s_g$ is positive. The characteristics of the range of ${{\cal I}}(g)$ are now used as a guide for a choice of domain for $H(g)$, namely the dense subspace $D(g_0,g_1)$ given by (\[D01\]). Lemma \[lemma2.2\] then asserts that $H(g)$ is symmetric on $D(g_0,g_1)$ for all $g$ in the parameter space $$\begin{gathered}
\label{tPi}
\tilde{\Pi}\equiv \left\{ g\in{{\mathbb R}}^4 \mid g_0,g_1>-1/2\right\}.\end{gathered}$$ The proofs of these lemmas involve technicalities that are not enlightening, so we have relegated them to Appendix \[appendix-A\].

Next, using the key identity (\[eiB\]), we show in Lemma \[lemma2.3\] that for all $g\in\tilde{\Pi}$ the subspace ${D(g_0,g_1)}$ is a core (domain of essential self-adjointness [@rs2]). This 