is true for $H(g)$, $g\in\tilde{\Pi}$. Furthermore, it is clear from the eigenvalues (\[teig\]) that the resolvent of $H_t(g_0,g_1)$ is compact. By the second resolvent identity this also holds true for the resolvent of $H(g)$, so the theorem follows.

This theorem also reveals the special character of the (at most four) self-adjoint extensions of $H(g)$ restricted to ${{\cal C}}$ (\[cC\]) that arise via ${{\cal I}}(g)$: They are the extensions whose trigonometric limits yield the (at most four) self-adjoint extensions of $H_t(g_0,g_1)$ restricted to ${{\cal C}}$ that are associated with the Jacobi polynomials. Of course, this is particularly clear when $V(g;x)$ and $V_t(g_0,g_1;x)$ vanish identically, cf. Section \[section4\].

For the remainder of this section we assume $g\in\Pi_r$. It follows from Theorems \[theorem2.4\] and \[theorem2.5\] that $H(g)$ has nondegenerate eigenvalues $$\begin{gathered}
E_0(g)<E_1(g)<E_2(g)<\cdots.\end{gathered}$$ Also, since $H(g)$ and $H_t(g_0,g_1)$ differ by a bounded self-adjoint operator (cf. (\[Vd\])), it follows from (\[teig\]) that the eigenvalues have asymptotics $$\begin{gathered}
E_m(g)\sim 4r^2m^2,\qquad m\to\infty.\end{gathered}$$ By Theorem \[theorem2.4\] the associated eigenvectors $u_m(g)$ are also eigenvectors of $T(g)$, and they are just the scalar multiples of $e_n(g)$ for some $n\in{{\mathbb N}}$. Hence we may and will fix the ordering choice of the $T(g)$-eigenvectors for a given degenerate eigenvalue (if any occur), 