${{\cal I}}(g), g\in\Pi$, has rank 1 when $s_g$ vanishes. By contrast, for $s_g>0$ the integral operator ${{\cal I}}(g)$ has dense range and trivial kernel: $$\begin{gathered}
\label{KerI}
{\rm Ker}( {{\cal I}}(g))=\{ 0\},\qquad {\rm Ran}({{\cal I}}(g))^{-}={{\cal H}},\qquad \forall \, g\in\Pi_r.\end{gathered}$$ This is clear from the above and the following lemma, which is proved in Appendix \[appendix-A\].

\[lemma2.1\] Letting $s_g>0$, suppose $\phi(y) \in L^1([0,\pi/2r],dy)$ satisfies $$\begin{gathered}
 \int_0^{\pi/2r}dy{{\cal S}}(g;x,y)\phi(y)=0.\end{gathered}$$ Then $\phi =0$.

To proceed, we note that the kernel function $\Psi(g;x,y)$ is real-valued. From (\[KerI\]), (\[cIad\]) and the HS property we then deduce that for a fixed $g\in\Pi_r$, there exist two ONBs $\{ e_n(g')\}_{n\in{{\mathbb N}}}$ and $\{ e_n(g)\}_{n\in{{\mathbb N}}}$ of real-valued functions such that $$\begin{gathered}
\label{Psirep}
\Psi(g;x,y)=\sum_{n=0}^{\infty}\nu_n(g)e_n(g;x)e_n(g';y),\qquad \nu_0\ge\nu_1\ge \cdots >0,\end{gathered}$$ with $\{ \nu_n\}\in l^2({{\mathbb N}})$. (The choice of the first ONB uniquely determines the second one; later on, we will return to the specific choice we make.) Moreover, since $e_n(g)$ belongs to the range of ${{\cal I}}(g)$, the above analysis implies that it is of the form $$\begin{gathered}
\label{ed}
e_n(g;x)=w(g;x)^{1/2}d_n(g;x),\end{gathered}$$ where $d_n(g;x)$ extends to a function that is analytic in $|\Im z|<\alpha/2$, even and $\pi/r$-periodic.

We are now prepared to deal with the Heun Hamiltonian $H(g)$. We choose as initial domain for $H(g)$ the 