  We classify all sets of the form
$\bigcup_{t\in\mathbb{R}}\mathrm{spec}(A+tB)$ where $A$ and $B$ are
self-adjoint operators and $B$ is bounded, non-negative, and non-zero. We show
that these sets are exactly the complements of discrete subsets of
$\mathbb{R}$, that is, of at most countable subsets of $\mathbb{R}$ that
contain none of their accumulation points.
