a corollary of Theorem \[theorem2.4\] that there exist distinct real numbers $\lambda_n(g)$ such that $$\begin{gathered}
\label{Hlam}
H(g)e_n(g)=\lambda_n(g)e_n(g), \qquad n\in{{\mathbb N}},\qquad g\in\Pi_r.\end{gathered}$$ At this stage, however, we have no other information on the $H(g)$-eigenvalues. In the following theorem, we obtain further results on the $H(g)$-spectrum that do not involve ${{\cal I}}(g)$, and which are valid for all $g\in\tilde{\Pi}$. As a preparation, note that $H(g)$ is obviously bounded below for $g_0,g_1\ge 1$. (Indeed, this entails that $V(g;x)$ is bounded below, cf. (\[Vg\]).) For $g_0$ and/or $g_1$ in $(0,1)$, however, $V(g;x)$ diverges to $-\infty$ as $x\downarrow 0$ and/or $x\uparrow \pi/2r$. Even so, our next theorem implies that $H(g)$ is still bounded below. Its proof is based on a comparison argument involving the trigonometric $BC_1$ Calogero–Moser Hamiltonian (\[Ht\]).

\[theorem2.5\] For all $g\in\tilde{\Pi}$, the spectrum of the operator $H(g)$ is a discrete set that is bounded below. Moreover, each eigenvalue has finite multiplicity.

The trigonometric and elliptic differential operators (\[Ht\]) and (\[Hg\]) differ by a real-valued potential $$\begin{gathered}
V_d(g;x)\equiv V_t(g_0,g_1;x)-V(g;x),\end{gathered}$$ that is bounded on $[0,\pi/2r]$. Hence we can associate to (\[Ht\]) the operator on ${D(g_0,g_1)}$ given by $$\begin{gathered}
\label{Vd}
H_t(g_0,g_1)\equiv H(g)+V_d(g;\cdot).\end{gathered}$$ This operator is essentially self-adjoint on ${D(g_0,g_1)}$, since $H(g)$ is e.s.a. on ${D(g_0,g_1)}$ by 