Interface asymptotics of Partial Bergman kernels around a critical level

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.