virtue of Lemma \[lemma2.3\] and the multiplication operator $V_d(g;\cdot)$ is bounded and self-adjoint. We now introduce the trigonometric weight function $$\begin{gathered}
w_t(g_0,g_1;x)\equiv (\sin rx)^{2g_0}(\cos rx)^{2g_1},\end{gathered}$$ and observe that we may rewrite ${D(g_0,g_1)}$ as $$\begin{gathered}
{D(g_0,g_1)}=w_t(g_0,g_1;x)^{1/2}{{\cal D}}.\end{gathered}$$ (Indeed, ${{\cal D}}$ (\[cD\]) is closed under products and we have $$\begin{gathered}
\left(\frac{s(x)}{\sin rx}\right)^{\lambda},\ \ \left(\frac{s(\pi/2r-x)}{\cos rx}\right)^{\mu}\in{{\cal D}},\qquad \lambda,\mu\in{{\mathbb R}},\end{gathered}$$ as is easily checked.)

Next, we calculate the similarity-transformed differential operator $$\begin{gathered}
A_t(g_0,g_1;x)     \equiv     w_t(g_0,g_1;x)^{-1/2}H_t(g_0,g_1;x)w_t(g_0,g_1;x)^{1/2}
\nonumber \\
 \phantom{A_t(g_0,g_1;x)}{}  =     -\frac{d^2}{dx^2}-2r(g_0\cot rx -g_1\tan rx)\frac{d}{dx}+r^2(g_0+g_1)^2.\end{gathered}$$ This yields an essentially self-adjoint operator $A_t(g_0,g_1)$ on the dense subspace ${{\cal D}}$ (given by (\[cD\])) of the Hilbert space $$\begin{gathered}
{{\cal H}}_t\equiv L^2([0,\pi/2r],w_t(g_0,g_1;x)dx).\end{gathered}$$ The crux is now that $A_t(g_0,g_1)$ has an orthogonal base of eigenvectors belonging to ${{\cal D}}$ with nonnegative eigenvalues, explicitly given by $$\begin{gathered}
\label{dmt}
d_{m,t}(g_0,g_1;x)= {}_2F_1(-m,g_0+g_1+m;g_0+1/2;\sin^2rx),\qquad m\in{{\mathbb N}},
\\
\label{teig}
A_t(g_0,g_1)d_{m,t}(g_0,g_1)=r^2(g_0+g_1+2m)^2d_{m,t}(g_0,g_1),\qquad m\in{{\mathbb N}}.\end{gathered}$$ Indeed, the r.h.s. of (\[dmt\]) yields polynomials of degree $m$ in $\sin^2rx$, and these polynomials amount to the Jacobi polynomials $$\begin{gathered}
P_m^{(\alpha,\beta)}(y),\qquad \alpha\equiv g_0-1/2,\qquad \beta\equiv g_1-1/2,\qquad y\equiv \cos 2rx,\qquad  m\in{{\mathbb N}},\end{gathered}$$ which form an orthogonal base in $$\begin{gathered}
{{\cal H}}_P\equiv L^2\big([-1,1],(1-y)^{\alpha}(1+y)^{\beta}dy\big),\end{gathered}$$ see for instance [@kosw].

From the above it follows that the spectrum of $H_t(g_0,g_1)$, $g_0,g_1>-1/2$, is bounded below, so by (\[Vd\]) the same 