prepares us for studying the self-adjoint closure (again denoted $H(g)$) by using ${{\cal I}}(g)$. Only a few simple steps are then needed to reach the conclusion that the above ONB $\{e_n(g)\}$ may be chosen so that it is an eigenvector ONB for $H(g)$. Moreover, since the eigenvectors belong to ${D(g_0,g_1)}$, it easily follows from the Frobenius method that the spectrum of $H(g)$ is simple, cf. Theorem \[theorem2.4\].

Whenever the potential $V(g;x)$ is bounded below, it is obvious that $H(g)$ is also bounded below. For $g_0$ and/or $g_1$ in $ (0,1)$, however, $V(g;x)$ is not bounded below, so an extra argument is needed to show that for the pertinent self-adjoint extensions of (\[Hg\]) semi-boundedness is preserved. We prove this by a comparison argument involving the trigonometric $BC_1$ Calogero–Moser Hamiltonian $$\begin{gathered}
H_t(g_0,g_1;x)\equiv -\frac{d^2}{dx^2}+V_t(g_0,g_1;x),\nonumber\\
V_t(g_0,g_1;x)\equiv \frac{r^2g_0(g_0-1)}{\sin^2rx}+\frac{r^2g_1(g_1-1)}{\cos^2rx}.\label{Ht}\end{gathered}$$ The point is that this operator is well defined and essentially self-adjoint on ${D(g_0,g_1)}$, and that it has an ONB of eigenvectors in ${D(g_0,g_1)}$ involving the well-known Jacobi polynomials, cf. Theorem \[theorem2.5\].

We would like to add that the trigonometric Hamiltonian (\[Ht\]) plays a key role in Takemura’s paper [@TakeHeun], where he studies the Heun Hamiltonian $H(g;x)$ (\[Hg\]) with $g_0,g_1\ge 1$ from a Hilbert space point of 