the eigenfunctions of $H(g)$ viewed as a differential operator, cf. e.g. the reviews . In recent years, this lore has been significantly enlarged by Takemura, whose work can be traced from his recent paper [@take].

Here, however, we are concerned with spectral properties of $H(g)$, reinterpreted as a self-adjoint operator on the Hilbert space $$\begin{gathered}
\label{cH}
{{\cal H}}\equiv L^2([0,\pi/2r],dx).\end{gathered}$$ A few of the properties we derive (such as the occurrence of solely discrete spectrum for suitably restricted $g$) follow from well-known arguments in the theory of Schrödinger operators [@rs2] (or, alternatively, from Sturm–Liouville theory and the theory of eigenfunction expansions). Our main tool for reobtaining known features and obtaining novel ones is the integral operator whose kernel is given by $$\begin{gathered}
\label{Psi}
 \Psi(g;x,y)\equiv w(g;x)^{1/2}{{\cal S}}(g;x,y)w(g';y)^{1/2},\qquad x,y\in(0,\pi/2r).
 \end{gathered}$$

This kernel and the definition of its ingredients can be found in I, cf. I Subsection 3.2. For ease of reference, however, we repeat the relevant definitions: $$\begin{gathered}
\label{defJN}
g'\equiv J_Ng,\qquad J_N\equiv
 \frac{1}{2} \left( \begin{array}{rrrr}
1  &  1  &  -1  &  1  \\
1  &  1  &  1  &  -1  \\
-1  &  1  &  1  &  1  \\
1  &  -1  &  1  &  1
\end{array} \right) ,
\\
\label{cS}
{{\cal S}}(g;x,y)\equiv 