  In a recent series of articles (arXiv:1604.06655, arXiv:1708.09267), the
authors have studied the transition behavior of partial Bergman kernels
$\Pi_{k, [E_1, E_2]}(z,w)$ and the associated DOS (density of states) $\Pi_{k,
[E_1, E_2]}(z)$ across the interface $\ccal$ between the allowed and forbidden
regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse
functions $H: M \to \R$ on a \kahler manifold. The allowed region is
$H^{-1}([E_1, E_2])$ and the interface $\ccal$ is its boundary. In prior
articles it was assumed that the endpoints $E_j$ were regular values of $H$.
This article completes the series by giving parallel results when an endpoint
is a critical value of $H$. In place of the Erf scaling asymptotics in a
$k^{-\half} $ tube around $\ccal$ for regular interfaces, one obtains
$\delta$-asymptotics in $k^{-\frac{1}{4}}$-tubes around singular points of a
critical interface. In $k^{-\half}$ tubes, the transition law is given by the
osculating metaplectic propagator.
