  Let $A(t)$ be a holomorphic family of self-adjoint operators of type (B) on a
complex Hilbert space $\mathcal{H}$. Kato-Rellich perturbation theory says that
isolated eigenvalues of $A(t)$ will be analytic functions of $t$ as long as
they remain below the minimum of the essential spectrum of $A(t)$. At a
threshold value $t_0$ where one of these eigenvalue functions hits the
essential spectrum, the corresponding point in the essential spectrum might or
might not be an eigenvalue of $A(t_0)$. Our results generalize a theorem of
Simon to give a sufficient condition for the minimum of the essential spectrum
to be an eigenvalue of $A(t_0)$ based on the rate at which eigenvalues approach
the essential spectrum. We also show that the rates at which the eigenvalues of
$A(t)$ can approach the essential spectrum from below correspond to eigenvalues
of a bounded self-adjoint operator. The key insight behind these results is the
essential numerical range which was recently extended to unbounded operators by
B\"{o}gli, Marletta, and Tretter.
