  For a system consisting of a quantum emitter coupled near threshold (band
edge) to a one-dimensional continuum with a van Hove singularity in the density
of states, we demonstrate general conditions such that a characteristic triple
level convergence occurs directly on the threshold as the coupling $g$ is shut
off. For small $g$ values the eigenvalue and norm of each of these states can
be expanded in a Puiseux expansion in terms of powers of $g^{2/3}$, which
suggests the influence of a third-order exceptional point. However, in the
actual $g \rightarrow 0$ limit, only two discrete states in fact coalesce as
the system can be reduced to a $2 \times 2$ Jordan block; the third state
instead merges with the continuum. Moreover, the decay width of the resonance
state involved in this convergence is significantly enhanced compared to the
usual Fermi golden rule, which is consistent with the Purcell effect. However,
non-Markovian dynamics due to the branch-point effect are also enhanced near
the threshold. Applying a perturbative analysis in terms of the Puiseux
expansion that takes into account the threshold influence, we show that the
combination of these effects results in quantum emitter decay of the unusual
form $1 - C t^{3/2}$ on the key timescale during which most of the decay
occurs. We then present two conditions that must be satisfied at the threshold
for the anomalous exceptional point to occur: the density of states must
contain an inverse square-root divergence and the potential must be
non-singular. We further show that when the energy of the quantum emitter is
detuned from threshold, the anomalous exceptional point splits into three
ordinary exceptional points, two of which appear in the complex-extended
parameter space. These results provide deeper insight into a well-known problem
in spontaneous decay at a photonic band edge.
