Interface asymptotics of Partial Bergman kernels around a critical level

In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels Π_k,[E1,E₂ ](z, w) and the associated DOS (density of states) Π_k,[E1,E₂ ](z) across the interface C between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions H:M →ℝ on a Kähler manifold. The allowed region is H⁻¹([E₁,E₂ ]) and the interface C 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⁻¹2 tube around C for regular interfaces, one obtains δ-asymptotics in k⁻¹4-tubes around singular points of a critical interface. In k⁻¹2 tubes, the transition law is given by the osculating metaplectic propagator.